O rosto é o círculo de equação geral dada abaixo (veja o seu número \(n\) nesta lista).

Ache a equação canônica deste círculo.

1. \(\displaystyle \quad x^{2} + 2 x + y^{2} + 4 y - 4 = 0\)

2. \(\displaystyle \quad x^{2} + 2 x + y^{2} + 4 y + 4 = 0\)

3. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 4 y - 1 = 0\)

4. \(\displaystyle \quad x^{2} - 2 x + y^{2} + 2 y + 1 = 0\)

5. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 2 y + 1 = 0\)

6. \(\displaystyle \quad x^{2} - 2 x + y^{2} - 2 y - 2 = 0\)

7. \(\displaystyle \quad x^{2} + 4 x + y^{2} + 4 y - 1 = 0\)

8. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 4 y - 8 = 0\)

9. \(\displaystyle \quad x^{2} + 4 x + y^{2} - 4 y + 7 = 0\)

10. \(\displaystyle \quad x^{2} + 2 x + y^{2} - 4 y - 4 = 0\)

11. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 2 y - 11 = 0\)

12. \(\displaystyle \quad x^{2} - 4 x + y^{2} + 4 y + 7 = 0\)

13. \(\displaystyle \quad x^{2} + 4 x + y^{2} - 2 y + 1 = 0\)

14. \(\displaystyle \quad x^{2} - 2 x + y^{2} - 2 y - 7 = 0\)

15. \(\displaystyle \quad x^{2} + 2 x + y^{2} - 2 y - 7 = 0\)

16. \(\displaystyle \quad x^{2} - 2 x + y^{2} + 4 y + 1 = 0\)

17. \(\displaystyle \quad x^{2} - 2 x + y^{2} - 2 y - 14 = 0\)

18. \(\displaystyle \quad x^{2} - 2 x + y^{2} + 4 y - 4 = 0\)

19. \(\displaystyle \quad x^{2} - 2 x + y^{2} - 4 y + 4 = 0\)

20. \(\displaystyle \quad x^{2} + 2 x + y^{2} + 2 y - 2 = 0\)

21. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 2 y - 4 = 0\)

22. \(\displaystyle \quad x^{2} + 2 x + y^{2} + 2 y - 7 = 0\)

23. \(\displaystyle \quad x^{2} - 4 x + y^{2} + 2 y - 4 = 0\)

24. \(\displaystyle \quad x^{2} + 2 x + y^{2} - 2 y - 2 = 0\)

25. \(\displaystyle \quad x^{2} + 4 x + y^{2} + 2 y - 4 = 0\)

26. \(\displaystyle \quad x^{2} + 4 x + y^{2} - 4 y - 1 = 0\)

27. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 4 y + 7 = 0\)

28. \(\displaystyle \quad x^{2} - 4 x + y^{2} + 4 y - 1 = 0\)

29. \(\displaystyle \quad x^{2} + 4 x + y^{2} - 2 y - 11 = 0\)

30. \(\displaystyle \quad x^{2} - 2 x + y^{2} - 2 y + 1 = 0\)

31. \(\displaystyle \quad x^{2} + 2 x + y^{2} + 2 y - 14 = 0\)

32. \(\displaystyle \quad x^{2} + 4 x + y^{2} + 2 y + 4 = 0\)

33. \(\displaystyle \quad x^{2} + 4 x + y^{2} - 2 y - 4 = 0\)

34. \(\displaystyle \quad x^{2} + 4 x + y^{2} + 4 y - 8 = 0\)

35. \(\displaystyle \quad x^{2} + 2 x + y^{2} - 2 y + 1 = 0\)

36. \(\displaystyle \quad x^{2} + 2 x + y^{2} - 4 y - 11 = 0\)

37. \(\displaystyle \quad x^{2} - 4 x + y^{2} - 4 y + 4 = 0\)

38. \(\displaystyle \quad x^{2} - 4 x + y^{2} + 2 y + 1 = 0\)

39. \(\displaystyle \quad x^{2} - 4 x + y^{2} + 4 y - 8 = 0\)

40. \(\displaystyle \quad x^{2} - 2 x + y^{2} + 2 y - 7 = 0\)

41. \(\displaystyle \quad x^{2} - 2 x + y^{2} + 2 y - 14 = 0\)

42. \(\displaystyle \quad x^{2} - 2 x + y^{2} - 4 y - 4 = 0\)

O nariz é a elipse de eixo maior horizontal com as coordenadas do centro, valores de \(a\) e de \(c\) dados abaixo (veja o seu número \(n\) nesta lista).

Ache a equação canônica desta elipse.

1. \(\displaystyle \quad \text{Centro} = \left( -1; \ -2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

2. \(\displaystyle \quad \text{Centro} = \left( -1; \ -2\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

3. \(\displaystyle \quad \text{Centro} = \left( 2; \ 2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

4. \(\displaystyle \quad \text{Centro} = \left( 1; \ -1\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

5. \(\displaystyle \quad \text{Centro} = \left( 2; \ 1\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

6. \(\displaystyle \quad \text{Centro} = \left( 1; \ 1\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

7. \(\displaystyle \quad \text{Centro} = \left( -2; \ -2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

8. \(\displaystyle \quad \text{Centro} = \left( 2; \ 2\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

9. \(\displaystyle \quad \text{Centro} = \left( -2; \ 2\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

10. \(\displaystyle \quad \text{Centro} = \left( -1; \ 2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

11. \(\displaystyle \quad \text{Centro} = \left( 2; \ 1\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

12. \(\displaystyle \quad \text{Centro} = \left( 2; \ -2\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

13. \(\displaystyle \quad \text{Centro} = \left( -2; \ 1\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

14. \(\displaystyle \quad \text{Centro} = \left( 1; \ 1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

15. \(\displaystyle \quad \text{Centro} = \left( -1; \ 1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

16. \(\displaystyle \quad \text{Centro} = \left( 1; \ -2\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

17. \(\displaystyle \quad \text{Centro} = \left( 1; \ 1\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

18. \(\displaystyle \quad \text{Centro} = \left( 1; \ -2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

19. \(\displaystyle \quad \text{Centro} = \left( 1; \ 2\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

20. \(\displaystyle \quad \text{Centro} = \left( -1; \ -1\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

21. \(\displaystyle \quad \text{Centro} = \left( 2; \ 1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

22. \(\displaystyle \quad \text{Centro} = \left( -1; \ -1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

23. \(\displaystyle \quad \text{Centro} = \left( 2; \ -1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

24. \(\displaystyle \quad \text{Centro} = \left( -1; \ 1\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

25. \(\displaystyle \quad \text{Centro} = \left( -2; \ -1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

26. \(\displaystyle \quad \text{Centro} = \left( -2; \ 2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

27. \(\displaystyle \quad \text{Centro} = \left( 2; \ 2\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

28. \(\displaystyle \quad \text{Centro} = \left( 2; \ -2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

29. \(\displaystyle \quad \text{Centro} = \left( -2; \ 1\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

30. \(\displaystyle \quad \text{Centro} = \left( 1; \ 1\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

31. \(\displaystyle \quad \text{Centro} = \left( -1; \ -1\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

32. \(\displaystyle \quad \text{Centro} = \left( -2; \ -1\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

33. \(\displaystyle \quad \text{Centro} = \left( -2; \ 1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

34. \(\displaystyle \quad \text{Centro} = \left( -2; \ -2\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

35. \(\displaystyle \quad \text{Centro} = \left( -1; \ 1\right)\ ,\quad a = \frac{11}{100}\ ,\quad c = \frac{1}{10}\)

36. \(\displaystyle \quad \text{Centro} = \left( -1; \ 2\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

37. \(\displaystyle \quad \text{Centro} = \left( 2; \ 2\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

38. \(\displaystyle \quad \text{Centro} = \left( 2; \ -1\right)\ ,\quad a = \frac{11}{50}\ ,\quad c = \frac{1}{5}\)

39. \(\displaystyle \quad \text{Centro} = \left( 2; \ -2\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

40. \(\displaystyle \quad \text{Centro} = \left( 1; \ -1\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

41. \(\displaystyle \quad \text{Centro} = \left( 1; \ -1\right)\ ,\quad a = \frac{11}{25}\ ,\quad c = \frac{2}{5}\)

42. \(\displaystyle \quad \text{Centro} = \left( 1; \ 2\right)\ ,\quad a = \frac{33}{100}\ ,\quad c = \frac{3}{10}\)

Os centros dos olhos são os focos da elipse com equação geral dada abaixo (veja o seu número \(n\) nesta lista).

Ache a equação canônica e as coordenadas dos focos desta elipse.

1. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} + \frac{242 y}{15} + \frac{2297}{240} = 0\)

2. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} + \frac{726 y}{35} + \frac{2169}{112} = 0\)

3. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{3146 y}{105} + \frac{67583}{1680} = 0\)

4. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} + \frac{968 y}{105} + \frac{7367}{1680} = 0\)

5. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{242 y}{15} + \frac{169}{12} = 0\)

6. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} - \frac{242 y}{15} + \frac{133}{12} = 0\)

7. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} + \frac{242 y}{15} + \frac{3017}{240} = 0\)

8. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{484 y}{15} + \frac{133}{3} = 0\)

9. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} - \frac{2662 y}{105} + \frac{53063}{1680} = 0\)

10. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} - \frac{3146 y}{105} + \frac{62543}{1680} = 0\)

11. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{726 y}{35} + \frac{624}{35} = 0\)

12. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} + \frac{726 y}{35} + \frac{2505}{112} = 0\)

13. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} - \frac{242 y}{15} + \frac{169}{12} = 0\)

14. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} - \frac{1936 y}{105} + \frac{21887}{1680} = 0\)

15. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} - \frac{1936 y}{105} + \frac{21887}{1680} = 0\)

16. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} + \frac{1936 y}{105} + \frac{6107}{420} = 0\)

17. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} - \frac{726 y}{35} + \frac{519}{35} = 0\)

18. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} + \frac{242 y}{15} + \frac{2297}{240} = 0\)

19. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} - \frac{2662 y}{105} + \frac{48023}{1680} = 0\)

20. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} + \frac{242 y}{35} + \frac{261}{140} = 0\)

21. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{1936 y}{105} + \frac{26927}{1680} = 0\)

22. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} + \frac{484 y}{105} - \frac{269}{336} = 0\)

23. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} + \frac{484 y}{105} + \frac{739}{336} = 0\)

24. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} - \frac{242 y}{15} + \frac{133}{12} = 0\)

25. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} + \frac{484 y}{105} + \frac{739}{336} = 0\)

26. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} - \frac{3146 y}{105} + \frac{67583}{1680} = 0\)

27. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{2662 y}{105} + \frac{53063}{1680} = 0\)

28. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} + \frac{242 y}{15} + \frac{3017}{240} = 0\)

29. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} - \frac{726 y}{35} + \frac{624}{35} = 0\)

30. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} - \frac{484 y}{35} + \frac{5037}{560} = 0\)

31. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} + \frac{242 y}{105} - \frac{379}{105} = 0\)

32. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} + \frac{968 y}{105} + \frac{12407}{1680} = 0\)

33. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} - \frac{1936 y}{105} + \frac{26927}{1680} = 0\)

34. \(\displaystyle \quad x^{2} + 4 x + \frac{121 y^{2}}{21} + \frac{484 y}{35} + \frac{261}{35} = 0\)

35. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} - \frac{484 y}{35} + \frac{5037}{560} = 0\)

36. \(\displaystyle \quad x^{2} + 2 x + \frac{121 y^{2}}{21} - \frac{484 y}{15} + \frac{124}{3} = 0\)

37. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} - \frac{968 y}{35} + \frac{5037}{140} = 0\)

38. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} + \frac{242 y}{35} + \frac{681}{140} = 0\)

39. \(\displaystyle \quad x^{2} - 4 x + \frac{121 y^{2}}{21} + \frac{484 y}{35} + \frac{261}{35} = 0\)

40. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} + \frac{484 y}{105} - \frac{269}{336} = 0\)

41. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} + \frac{242 y}{105} - \frac{379}{105} = 0\)

42. \(\displaystyle \quad x^{2} - 2 x + \frac{121 y^{2}}{21} - \frac{3146 y}{105} + \frac{62543}{1680} = 0\)

Cada olho é uma elipse de eixo maior vertical com os valores de \(a\) e de \(c\) dados abaixo (veja o seu número \(n\) nesta lista).

Ache as equações gerais destas elipses.

1. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

2. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

3. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

4. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

5. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

6. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

7. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

8. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

9. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

10. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

11. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

12. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

13. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

14. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

15. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

16. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

17. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

18. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

19. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

20. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

21. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

22. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

23. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

24. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

25. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

26. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

27. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

28. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

29. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

30. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

31. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

32. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

33. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

34. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

35. \(\displaystyle \quad a = \frac{1}{3}\ ,\quad c = \frac{1}{4}\)

36. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

37. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

38. \(\displaystyle \quad a = \frac{2}{3}\ ,\quad c = \frac{1}{2}\)

39. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

40. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

41. \(\displaystyle \quad a = \frac{4}{3}\ ,\quad c = 1\)

42. \(\displaystyle \quad a = 1\ ,\quad c = \frac{3}{4}\)

A parte superior da boca é uma parábola com concavidade para cima, com as coordenadas do foco \(F\) e a distância focal \(p\) dadas abaixo (veja o seu número \(n\) nesta lista).

Ache a equação canônica desta parábola.

1. \(\displaystyle \quad F = \left( -1; \ -3\right)\ ,\quad p = \frac{3}{4}\)

2. \(\displaystyle \quad F = \left( -1; \ - \frac{7}{3}\right)\ ,\quad p = \frac{1}{4}\)

3. \(\displaystyle \quad F = \left( 2; \ 1\right)\ ,\quad p = \frac{3}{4}\)

4. \(\displaystyle \quad F = \left( 1; \ - \frac{4}{3}\right)\ ,\quad p = \frac{1}{4}\)

5. \(\displaystyle \quad F = \left( 2; \ \frac{1}{3}\right)\ ,\quad p = \frac{1}{2}\)

6. \(\displaystyle \quad F = \left( 1; \ \frac{1}{3}\right)\ ,\quad p = \frac{1}{2}\)

7. \(\displaystyle \quad F = \left( -2; \ -3\right)\ ,\quad p = \frac{3}{4}\)

8. \(\displaystyle \quad F = \left( 2; \ \frac{2}{3}\right)\ ,\quad p = 1\)

9. \(\displaystyle \quad F = \left( -2; \ \frac{5}{3}\right)\ ,\quad p = \frac{1}{4}\)

10. \(\displaystyle \quad F = \left( -1; \ 1\right)\ ,\quad p = \frac{3}{4}\)

11. \(\displaystyle \quad F = \left( 2; \ - \frac{1}{3}\right)\ ,\quad p = 1\)

12. \(\displaystyle \quad F = \left( 2; \ - \frac{7}{3}\right)\ ,\quad p = \frac{1}{4}\)

13. \(\displaystyle \quad F = \left( -2; \ \frac{1}{3}\right)\ ,\quad p = \frac{1}{2}\)

14. \(\displaystyle \quad F = \left( 1; \ 0\right)\ ,\quad p = \frac{3}{4}\)

15. \(\displaystyle \quad F = \left( -1; \ 0\right)\ ,\quad p = \frac{3}{4}\)

16. \(\displaystyle \quad F = \left( 1; \ - \frac{8}{3}\right)\ ,\quad p = \frac{1}{2}\)

17. \(\displaystyle \quad F = \left( 1; \ - \frac{1}{3}\right)\ ,\quad p = 1\)

18. \(\displaystyle \quad F = \left( 1; \ -3\right)\ ,\quad p = \frac{3}{4}\)

19. \(\displaystyle \quad F = \left( 1; \ \frac{5}{3}\right)\ ,\quad p = \frac{1}{4}\)

20. \(\displaystyle \quad F = \left( -1; \ - \frac{5}{3}\right)\ ,\quad p = \frac{1}{2}\)

21. \(\displaystyle \quad F = \left( 2; \ 0\right)\ ,\quad p = \frac{3}{4}\)

22. \(\displaystyle \quad F = \left( -1; \ -2\right)\ ,\quad p = \frac{3}{4}\)

23. \(\displaystyle \quad F = \left( 2; \ -2\right)\ ,\quad p = \frac{3}{4}\)

24. \(\displaystyle \quad F = \left( -1; \ \frac{1}{3}\right)\ ,\quad p = \frac{1}{2}\)

25. \(\displaystyle \quad F = \left( -2; \ -2\right)\ ,\quad p = \frac{3}{4}\)

26. \(\displaystyle \quad F = \left( -2; \ 1\right)\ ,\quad p = \frac{3}{4}\)

27. \(\displaystyle \quad F = \left( 2; \ \frac{5}{3}\right)\ ,\quad p = \frac{1}{4}\)

28. \(\displaystyle \quad F = \left( 2; \ -3\right)\ ,\quad p = \frac{3}{4}\)

29. \(\displaystyle \quad F = \left( -2; \ - \frac{1}{3}\right)\ ,\quad p = 1\)

30. \(\displaystyle \quad F = \left( 1; \ \frac{2}{3}\right)\ ,\quad p = \frac{1}{4}\)

31. \(\displaystyle \quad F = \left( -1; \ - \frac{7}{3}\right)\ ,\quad p = 1\)

32. \(\displaystyle \quad F = \left( -2; \ - \frac{4}{3}\right)\ ,\quad p = \frac{1}{4}\)

33. \(\displaystyle \quad F = \left( -2; \ 0\right)\ ,\quad p = \frac{3}{4}\)

34. \(\displaystyle \quad F = \left( -2; \ - \frac{10}{3}\right)\ ,\quad p = 1\)

35. \(\displaystyle \quad F = \left( -1; \ \frac{2}{3}\right)\ ,\quad p = \frac{1}{4}\)

36. \(\displaystyle \quad F = \left( -1; \ \frac{2}{3}\right)\ ,\quad p = 1\)

37. \(\displaystyle \quad F = \left( 2; \ \frac{4}{3}\right)\ ,\quad p = \frac{1}{2}\)

38. \(\displaystyle \quad F = \left( 2; \ - \frac{5}{3}\right)\ ,\quad p = \frac{1}{2}\)

39. \(\displaystyle \quad F = \left( 2; \ - \frac{10}{3}\right)\ ,\quad p = 1\)

40. \(\displaystyle \quad F = \left( 1; \ -2\right)\ ,\quad p = \frac{3}{4}\)

41. \(\displaystyle \quad F = \left( 1; \ - \frac{7}{3}\right)\ ,\quad p = 1\)

42. \(\displaystyle \quad F = \left( 1; \ 1\right)\ ,\quad p = \frac{3}{4}\)

A parte inferior da boca também é uma parábola, cuja equação geral é dada abaixo (veja o seu número \(n\) nesta lista).

Ache a equação canônica desta parábola.

1. \(\displaystyle \quad x^{2} + 2 x - \frac{27 y}{13} - \frac{217}{26} = 0\)

2. \(\displaystyle \quad x^{2} + 2 x - \frac{9 y}{13} - \frac{25}{26} = 0\)

3. \(\displaystyle \quad x^{2} - 4 x - \frac{27 y}{13} + \frac{77}{26} = 0\)

4. \(\displaystyle \quad x^{2} - 2 x - \frac{9 y}{13} - \frac{7}{26} = 0\)

5. \(\displaystyle \quad x^{2} - 4 x - \frac{18 y}{13} + \frac{40}{13} = 0\)

6. \(\displaystyle \quad x^{2} - 2 x - \frac{18 y}{13} + \frac{1}{13} = 0\)

7. \(\displaystyle \quad x^{2} + 4 x - \frac{27 y}{13} - \frac{139}{26} = 0\)

8. \(\displaystyle \quad x^{2} - 4 x - \frac{36 y}{13} + \frac{4}{13} = 0\)

9. \(\displaystyle \quad x^{2} + 4 x - \frac{9 y}{13} + \frac{125}{26} = 0\)

10. \(\displaystyle \quad x^{2} + 2 x - \frac{27 y}{13} - \frac{1}{26} = 0\)

11. \(\displaystyle \quad x^{2} - 4 x - \frac{36 y}{13} - \frac{32}{13} = 0\)

12. \(\displaystyle \quad x^{2} - 4 x - \frac{9 y}{13} + \frac{53}{26} = 0\)

13. \(\displaystyle \quad x^{2} + 4 x - \frac{18 y}{13} + \frac{40}{13} = 0\)

14. \(\displaystyle \quad x^{2} - 2 x - \frac{27 y}{13} - \frac{55}{26} = 0\)

15. \(\displaystyle \quad x^{2} + 2 x - \frac{27 y}{13} - \frac{55}{26} = 0\)

16. \(\displaystyle \quad x^{2} - 2 x - \frac{18 y}{13} - \frac{53}{13} = 0\)

17. \(\displaystyle \quad x^{2} - 2 x - \frac{36 y}{13} - \frac{71}{13} = 0\)

18. \(\displaystyle \quad x^{2} - 2 x - \frac{27 y}{13} - \frac{217}{26} = 0\)

19. \(\displaystyle \quad x^{2} - 2 x - \frac{9 y}{13} + \frac{47}{26} = 0\)

20. \(\displaystyle \quad x^{2} + 2 x - \frac{18 y}{13} - \frac{35}{13} = 0\)

21. \(\displaystyle \quad x^{2} - 4 x - \frac{27 y}{13} + \frac{23}{26} = 0\)

22. \(\displaystyle \quad x^{2} + 2 x - \frac{27 y}{13} - \frac{163}{26} = 0\)

23. \(\displaystyle \quad x^{2} - 4 x - \frac{27 y}{13} - \frac{85}{26} = 0\)

24. \(\displaystyle \quad x^{2} + 2 x - \frac{18 y}{13} + \frac{1}{13} = 0\)

25. \(\displaystyle \quad x^{2} + 4 x - \frac{27 y}{13} - \frac{85}{26} = 0\)

26. \(\displaystyle \quad x^{2} + 4 x - \frac{27 y}{13} + \frac{77}{26} = 0\)

27. \(\displaystyle \quad x^{2} - 4 x - \frac{9 y}{13} + \frac{125}{26} = 0\)

28. \(\displaystyle \quad x^{2} - 4 x - \frac{27 y}{13} - \frac{139}{26} = 0\)

29. \(\displaystyle \quad x^{2} + 4 x - \frac{36 y}{13} - \frac{32}{13} = 0\)

30. \(\displaystyle \quad x^{2} - 2 x - \frac{9 y}{13} + \frac{29}{26} = 0\)

31. \(\displaystyle \quad x^{2} + 2 x - \frac{36 y}{13} - 11 = 0\)

32. \(\displaystyle \quad x^{2} + 4 x - \frac{9 y}{13} + \frac{71}{26} = 0\)

33. \(\displaystyle \quad x^{2} + 4 x - \frac{27 y}{13} + \frac{23}{26} = 0\)

34. \(\displaystyle \quad x^{2} + 4 x - \frac{36 y}{13} - \frac{140}{13} = 0\)

35. \(\displaystyle \quad x^{2} + 2 x - \frac{9 y}{13} + \frac{29}{26} = 0\)

36. \(\displaystyle \quad x^{2} + 2 x - \frac{36 y}{13} - \frac{35}{13} = 0\)

37. \(\displaystyle \quad x^{2} - 4 x - \frac{18 y}{13} + \frac{58}{13} = 0\)

38. \(\displaystyle \quad x^{2} - 4 x - \frac{18 y}{13} + \frac{4}{13} = 0\)

39. \(\displaystyle \quad x^{2} - 4 x - \frac{36 y}{13} - \frac{140}{13} = 0\)

40. \(\displaystyle \quad x^{2} - 2 x - \frac{27 y}{13} - \frac{163}{26} = 0\)

41. \(\displaystyle \quad x^{2} - 2 x - \frac{36 y}{13} - 11 = 0\)

42. \(\displaystyle \quad x^{2} - 2 x - \frac{27 y}{13} - \frac{1}{26} = 0\)

Escreva uma inequação que represente a região da boca que está preenchida em vermelho na figura.

No Geogebra, entre a inequação e configure-a para que a área seja preenchida em vermelho.

As orelhas são os ramos de uma hipérbole de eixo real horizontal, de excentricidade \(11/10\), cujos vértices são os dois pontos do círculo do rosto que têm a coordenada \(y\) igual ao valor abaixo (veja o seu número \(n\) nesta lista).

1. \(\displaystyle \quad y = -2\)

2. \(\displaystyle \quad y = -2\)

3. \(\displaystyle \quad y = 2\)

4. \(\displaystyle \quad y = -1\)

5. \(\displaystyle \quad y = 1\)

6. \(\displaystyle \quad y = 1\)

7. \(\displaystyle \quad y = -2\)

8. \(\displaystyle \quad y = 2\)

9. \(\displaystyle \quad y = 2\)

10. \(\displaystyle \quad y = 2\)

11. \(\displaystyle \quad y = 1\)

12. \(\displaystyle \quad y = -2\)

13. \(\displaystyle \quad y = 1\)

14. \(\displaystyle \quad y = 1\)

15. \(\displaystyle \quad y = 1\)

16. \(\displaystyle \quad y = -2\)

17. \(\displaystyle \quad y = 1\)

18. \(\displaystyle \quad y = -2\)

19. \(\displaystyle \quad y = 2\)

20. \(\displaystyle \quad y = -1\)

21. \(\displaystyle \quad y = 1\)

22. \(\displaystyle \quad y = -1\)

23. \(\displaystyle \quad y = -1\)

24. \(\displaystyle \quad y = 1\)

25. \(\displaystyle \quad y = -1\)

26. \(\displaystyle \quad y = 2\)

27. \(\displaystyle \quad y = 2\)

28. \(\displaystyle \quad y = -2\)

29. \(\displaystyle \quad y = 1\)

30. \(\displaystyle \quad y = 1\)

31. \(\displaystyle \quad y = -1\)

32. \(\displaystyle \quad y = -1\)

33. \(\displaystyle \quad y = 1\)

34. \(\displaystyle \quad y = -2\)

35. \(\displaystyle \quad y = 1\)

36. \(\displaystyle \quad y = 2\)

37. \(\displaystyle \quad y = 2\)

38. \(\displaystyle \quad y = -1\)

39. \(\displaystyle \quad y = -2\)

40. \(\displaystyle \quad y = -1\)

41. \(\displaystyle \quad y = -1\)

42. \(\displaystyle \quad y = 2\)

Ache a equação canônica desta hipérbole.

Escreva uma inequação que represente a região das orelhas que está preenchida em rosa na figura. A distância \(d\) entre o centro do rosto e a borda vertical de cada orelha é a dada abaixo (veja o seu número \(n\) nesta lista).

1. \(\displaystyle \quad d = \frac{27}{8}\)

2. \(\displaystyle \quad d = \frac{9}{8}\)

3. \(\displaystyle \quad d = \frac{27}{8}\)

4. \(\displaystyle \quad d = \frac{9}{8}\)

5. \(\displaystyle \quad d = \frac{9}{4}\)

6. \(\displaystyle \quad d = \frac{9}{4}\)

7. \(\displaystyle \quad d = \frac{27}{8}\)

8. \(\displaystyle \quad d = \frac{9}{2}\)

9. \(\displaystyle \quad d = \frac{9}{8}\)

10. \(\displaystyle \quad d = \frac{27}{8}\)

11. \(\displaystyle \quad d = \frac{9}{2}\)

12. \(\displaystyle \quad d = \frac{9}{8}\)

13. \(\displaystyle \quad d = \frac{9}{4}\)

14. \(\displaystyle \quad d = \frac{27}{8}\)

15. \(\displaystyle \quad d = \frac{27}{8}\)

16. \(\displaystyle \quad d = \frac{9}{4}\)

17. \(\displaystyle \quad d = \frac{9}{2}\)

18. \(\displaystyle \quad d = \frac{27}{8}\)

19. \(\displaystyle \quad d = \frac{9}{8}\)

20. \(\displaystyle \quad d = \frac{9}{4}\)

21. \(\displaystyle \quad d = \frac{27}{8}\)

22. \(\displaystyle \quad d = \frac{27}{8}\)

23. \(\displaystyle \quad d = \frac{27}{8}\)

24. \(\displaystyle \quad d = \frac{9}{4}\)

25. \(\displaystyle \quad d = \frac{27}{8}\)

26. \(\displaystyle \quad d = \frac{27}{8}\)

27. \(\displaystyle \quad d = \frac{9}{8}\)

28. \(\displaystyle \quad d = \frac{27}{8}\)

29. \(\displaystyle \quad d = \frac{9}{2}\)

30. \(\displaystyle \quad d = \frac{9}{8}\)

31. \(\displaystyle \quad d = \frac{9}{2}\)

32. \(\displaystyle \quad d = \frac{9}{8}\)

33. \(\displaystyle \quad d = \frac{27}{8}\)

34. \(\displaystyle \quad d = \frac{9}{2}\)

35. \(\displaystyle \quad d = \frac{9}{8}\)

36. \(\displaystyle \quad d = \frac{9}{2}\)

37. \(\displaystyle \quad d = \frac{9}{4}\)

38. \(\displaystyle \quad d = \frac{9}{4}\)

39. \(\displaystyle \quad d = \frac{9}{2}\)

40. \(\displaystyle \quad d = \frac{27}{8}\)

41. \(\displaystyle \quad d = \frac{9}{2}\)

42. \(\displaystyle \quad d = \frac{27}{8}\)

No Geogebra, entre a inequação e configure-a para que a área seja preenchida em rosa.