Conic Sections - AP Precalculus

1. Find the center of the circle $(x-3)^2 + (y+5)^2 = 36$.

   (3, -5). The center is at $(h,k)$ in the vertex form $(x-h)^2 + (y-k)^2 = r^2$.

2. Identify the conic section: $x^2 - y^2 = 16$.

   Hyperbola. Difference of squares with equal coefficients indicates a hyperbola.

3. Identify the conic section: $y^2 = 4x$.

   Parabola. Square term on $y$ with linear $x$ indicates a horizontal parabola.

4. What is the formula for the latus rectum of a parabola?

   $|4p|$. The chord through the focus perpendicular to the axis of symmetry.

5. State the formula for the area of an ellipse with semi-axes $a$ and $b$.

   $A = \pi ab$. Where $a$ and $b$ are the lengths of the semi-major and semi-minor axes.

6. Identify the conic section: $x^2 + 4x + y^2 - 6y = 12$.

   Circle. Complete the square to verify equal coefficients for $x^2$ and $y^2$.

7. Identify the conic section: $x^2 + y^2 - 4x + 6y = 12$.

   Circle. Complete the square to verify it has equal coefficients for $x^2$ and $y^2$ terms.

8. Find the foci of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$.

   Foci: $(\pm 5, 0)$. From $c^2 = a^2 + b^2 = 9 + 16 = 25$, so $c = 5$.

9. Which conic section is represented by $4x^2 - 9y^2 = 36$?

   Hyperbola. Divide by 36 to get standard form with $a^2 = 9$ and $b^2 = 4$.

10. What is the relationship between $a$, $b$, and $c$ in a hyperbola?

    $c^2 = a^2 + b^2$. For hyperbolas, $c > a$ since the foci are outside the vertices.

11. What is the definition of a parabola?

    A set of points equidistant from a point (focus) and a line (directrix). This property defines the parabola's unique geometric shape.

12. Identify the conic section: $x^2 + 6x + y^2 - 4y = 0$.

    Circle. Complete the square to verify it forms a circle equation.

13. What is the definition of an ellipse?

    A set of points where the sum of distances to two foci is constant. The sum equals $2a$, where $a$ is the semi-major axis length.

14. Identify the conic section: $9x^2 + 16y^2 = 144$.

    Ellipse. Divide by 144 to get standard form with positive coefficients.

15. Which conic section has an eccentricity $e > 1$?

    Hyperbola. Eccentricity greater than 1 distinguishes hyperbolas from other conics.

16. Identify the conic section: $25x^2 + 25y^2 = 625$.

    Circle. Equal coefficients (25) for $x^2$ and $y^2$ terms indicate a circle.

17. Which conic section is represented by $x^2 - y^2 = 1$?

    Hyperbola. Difference of squares form indicates a hyperbola with $a^2 = b^2 = 1$.

18. What is the standard form of a hyperbola centered at origin?

    $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Subtraction between squared terms indicates a hyperbola opening horizontally.

19. Find the focus of the parabola $y^2 = 16x$.

    (4, 0). From $y^2 = 16x$, we get $4p = 16$, so $p = 4$ and focus is at $(p,0)$.

20. Find the equation of the directrix of the parabola $x^2 = 8y$.

    $y = -2$. From $x^2 = 8y$, we have $4p = 8$, so $p = 2$ and directrix is $y = -p$.

21. State the formula for the eccentricity of an ellipse.

    $e = \frac{c}{a}$ where $c = \sqrt{a^2 - b^2}$. For ellipses, $0 < e < 1$ since $c < a$ always.

22. State the formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.

    $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Derived from the Pythagorean theorem in coordinate geometry.

23. State the general form of a conic section equation.

    $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. The discriminant $B^2 - 4AC$ determines the conic type.

24. What is the definition of a hyperbola?

    A set of points where the difference of distances to two foci is constant. The absolute value of the difference equals $2a$, where $a$ is the semi-major axis.

25. What is the directrix of a parabola?

    A fixed line used to define the parabola. Points on the parabola are equidistant from focus and directrix.

26. What is the definition of a circle?

    A set of points equidistant from a fixed point (center). All points are the same distance from the center point.

27. What is the formula for the semi-major axis of an ellipse?

    $a$ in $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The larger denominator corresponds to the major axis length.

28. Identify the conic section: $x^2 + 2x + y^2 + 4y = 1$.

    Circle. Complete the square to verify it has equal coefficients for squared terms.

29. What is the standard form of an ellipse centered at origin?

    $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Where $a$ and $b$ are the semi-major and semi-minor axis lengths.

30. What is the equation for the asymptotes of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

    $y = \pm \frac{b}{a}x$. The slopes are $\pm\frac{b}{a}$ for the standard hyperbola form.

31. What is the standard form equation of a circle centered at origin?

    $x^2 + y^2 = r^2$. Where $r$ is the radius from the center at $(0,0)$.

32. Find the center and radii lengths of the ellipse $9x^2 + 4y^2 = 36$.

    Center: (0,0), $a=2$, $b=3$. Divide by 36: $\frac{x^2}{4} + \frac{y^2}{9} = 1$, so $a^2 = 4$, $b^2 = 9$.

33. What is the formula for the eccentricity of a conic section?

    $e = \frac{c}{a}$. Where $c$ is the focal distance and $a$ is the semi-major axis.

34. What is the formula for the focal distance of an ellipse?

    $c = \sqrt{a^2 - b^2}$. For an ellipse, $c < a$ since the foci are inside the ellipse.

35. What is the equation of a circle with center $(h,k)$ and radius $r$?

    $(x-h)^2 + (y-k)^2 = r^2$. Standard form for any circle with center and radius specified.

36. Find the vertex of the parabola $y = 3(x-2)^2 + 5$.

    (2, 5). The vertex form shows the vertex at $(h,k)$ where the parabola turns.

37. What is the standard form equation of a vertical parabola?

    $x^2 = 4py$. Where $p$ is the distance from vertex to focus and directrix.

38. What is the equation for the asymptotes of a hyperbola centered at origin?

    $y = \pm \frac{b}{a}x$. Lines the hyperbola approaches as $x$ and $y$ approach infinity.

39. What is the equation for a parabola with vertex $(h, k)$ and focus $(h, k+p)$?

    $(x-h)^2 = 4p(y-k)$. Vertex form for vertical parabolas with vertex at $(h,k)$.

40. What is the standard form equation of a parabola with vertex at origin?

    $y = ax^2$. Where $a$ determines the width and opens vertically.

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