AP Precalculus Flashcards: Conic Sections

What this deck covers

This deck focuses on Conic Sections, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: What is the formula for the semi-major axis of an ellipse?

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

Flashcard 3: Identify the conic section: $x^2 - y^2 = 16$.

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

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

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

Flashcard 5: What is the formula for the latus rectum of a parabola?

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

Flashcard 6: State the formula for the area of an ellipse with semi-axes $a$ and $b$.

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

Flashcard 7: What is the formula for the focal distance of an ellipse?

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

Flashcard 8: Identify the conic section: $x^2 + 4x + y^2 - 6y = 12$.

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

Flashcard 9: Identify the conic section: $x^2 + y^2 - 4x + 6y = 12$.

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

Flashcard 10: Find the foci of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$.

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

Flashcard 11: Which conic section is represented by $4x^2 - 9y^2 = 36$?

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

Flashcard 12: What is the definition of a circle?

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

Flashcard 13: Identify the conic section: $x^2 + 2x + y^2 + 4y = 1$.

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

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

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

Flashcard 15: What is the relationship between $a$, $b$, and $c$ in a hyperbola?

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

Flashcard 16: What is the definition of a parabola?

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

Flashcard 17: State the formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.

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

Flashcard 18: Find the focus of the parabola $y^2 = 16x$.

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

Flashcard 19: Identify the conic section: $x^2 + 6x + y^2 - 4y = 0$.

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

Flashcard 20: What is the definition of an ellipse?

Answer: 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.

Flashcard 21: What is the definition of an ellipse?

Answer: 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.

Flashcard 22: Identify the conic section: $9x^2 + 16y^2 = 144$.

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

Flashcard 23: What is the formula for the latus rectum of a parabola?

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

Flashcard 24: What is the standard form of an ellipse centered at origin?

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

Flashcard 25: Find the vertex of the parabola $y = 3(x-2)^2 + 5$.

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

Flashcard 26: Which conic section has an eccentricity $e > 1$?

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

Flashcard 27: Identify the conic section: $25x^2 + 25y^2 = 625$.

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

Flashcard 28: Which conic section is represented by $x^2 - y^2 = 1$?

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

Flashcard 29: Find the center of the circle $(x-3)^2 + (y+5)^2 = 36$.

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

Flashcard 30: What is the standard form of a hyperbola centered at origin?

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

Flashcard 31: State the general form of a conic section equation.

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

Flashcard 32: What is the definition of a hyperbola?

Answer: 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.

Flashcard 33: Find the foci of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$.

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

Flashcard 34: State the formula for the area of an ellipse with semi-axes $a$ and $b$.

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

Flashcard 35: What is the standard form of a hyperbola centered at origin?

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

Flashcard 36: Identify the conic section: $x^2 + y^2 - 4x + 6y = 12$.

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

Flashcard 37: Find the focus of the parabola $y^2 = 16x$.

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

Flashcard 38: Identify the conic section: $x^2 + 6x + y^2 - 4y = 0$.

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

Flashcard 39: Find the equation of the directrix of the parabola $x^2 = 8y$.

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

Flashcard 40: Identify the conic section: $y^2 = 4x$.

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

Flashcard 41: State the formula for the eccentricity of an ellipse.

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

Flashcard 42: Which conic section is represented by $4x^2 - 9y^2 = 36$?

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

Flashcard 43: State the formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.

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

Flashcard 44: What is the relationship between $a$, $b$, and $c$ in a hyperbola?

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

Flashcard 45: What is the standard form equation of a vertical parabola?

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

Flashcard 46: Find the equation of the directrix of the parabola $x^2 = 8y$.

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

Flashcard 47: What is the standard form equation of a parabola with vertex at origin?

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

Flashcard 48: State the general form of a conic section equation.

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

Flashcard 49: What is the definition of a hyperbola?

Answer: 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.

Flashcard 50: Which conic section is represented by $x^2 - y^2 = 1$?

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

Flashcard 51: What is the equation of a circle with center $(h,k)$ and radius $r$?

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

Flashcard 52: Identify the conic section: $x^2 + 4x + y^2 - 6y = 12$.

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

Flashcard 53: What is the directrix of a parabola?

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

Flashcard 54: State the formula for the eccentricity of an ellipse.

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

Flashcard 55: What is the definition of a circle?

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

Flashcard 56: What is the equation for the asymptotes of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

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

Flashcard 57: Identify the conic section: $x^2 - y^2 = 16$.

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

Flashcard 58: What is the definition of a parabola?

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

Flashcard 59: What is the formula for the semi-major axis of an ellipse?

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

Flashcard 60: What is the equation for a parabola with vertex $(h, k)$ and focus $(h, k+p)$?

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

Flashcard 61: Identify the conic section: $x^2 + 2x + y^2 + 4y = 1$.

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

Flashcard 62: Find the center and radii lengths of the ellipse $9x^2 + 4y^2 = 36$.

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

Flashcard 63: What is the directrix of a parabola?

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

Flashcard 64: What is the standard form of an ellipse centered at origin?

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

Flashcard 65: What is the equation for the asymptotes of a hyperbola centered at origin?

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

Flashcard 66: What is the equation for the asymptotes of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

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

Flashcard 67: What is the standard form equation of a circle centered at origin?

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

Flashcard 68: What is the standard form equation of a circle centered at origin?

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

Flashcard 69: Find the center and radii lengths of the ellipse $9x^2 + 4y^2 = 36$.

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

Flashcard 70: What is the formula for the eccentricity of a conic section?

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

Flashcard 71: What is the formula for the focal distance of an ellipse?

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

Flashcard 72: What is the equation of a circle with center $(h,k)$ and radius $r$?

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

Flashcard 73: Find the vertex of the parabola $y = 3(x-2)^2 + 5$.

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

Flashcard 74: Identify the conic section: $25x^2 + 25y^2 = 625$.

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

Flashcard 75: What is the standard form equation of a vertical parabola?

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

Flashcard 76: What is the formula for the eccentricity of a conic section?

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

Flashcard 77: What is the equation for the asymptotes of a hyperbola centered at origin?

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

Flashcard 78: Which conic section has an eccentricity $e > 1$?

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

Flashcard 79: What is the equation for a parabola with vertex $(h, k)$ and focus $(h, k+p)$?

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

Flashcard 80: What is the standard form equation of a parabola with vertex at origin?

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