Conic Sections

If $\alpha = 50^{\circ}$ and $\beta = 50^{\circ}$, and the plane does not pass through the vertex, we get a:

A point is a degenerate case of which conic section?

If a plane is parallel to the generator of a cone, the angle $\beta$ is:

Which component of the cone is a straight line that moves while staying fixed at the vertex?

When a plane intersects a double cone, the section is a hyperbola. This hyperbola consists of how many disjoint curves?

If a cone has a semi-vertical angle $\alpha = 30^\circ$, which of the following $\beta$ values would result in a parabola?

Which of the following conic sections is the only one that is a closed curve?

A hyperbola is formed when $0 \le \beta < \alpha$. If $\beta$ is very close to $\alpha$, the hyperbola resembles which of the following near its vertex?

Consider a plane $P$ intersecting a cone. If $P$ is moved parallel to itself, which property of the conic section remains unchanged?

If the semi-vertical angle $\alpha$ of a cone increases, while the plane angle $\beta$ is kept constant such that $\beta > \alpha$, the eccentricity of the resulting ellipse:

The equation of a circle with diameter endpoints $(-1, 0)$ and $(1, 0)$ is:

If the equation $x^2 + y^2 + 4x - 6y + \\lambda = 0$ represents a circle of radius $4$, find $\\lambda$.

The center and radius of the circle $x^2 + y^2 = 20$ are:

What is the equation of the circle whose center is $(0, 0)$ and which passes through the point $(-3, -4)$?

If the radius of the circle $x^2 + y^2 - 4x - 6y + k = 0$ is $5$, find $k$.

Find the equation of the circle with center at $(1, 2)$ and area $16\pi$.

Determine the diameter of the circle $x^2 + y^2 - 2x - 4y - 4 = 0$.

Two circles $x^2 + y^2 = a^2$ and $(x - c)^2 + y^2 = b^2$ touch each other externally. Which of the following is true?

The area of the circle that is centered at $(1, 2)$ and passes through the point $(4, 6)$ is:

If the circles $x^2 + y^2 = r^2$ and $x^2 + y^2 - 10x + 16 = 0$ intersect at two distinct points, then the radius $r$ satisfies:

What is the axis of symmetry for the parabola $y^2 = 18x$?

What is the length of the latus rectum for the parabola $x^2 = y$?

The vertex of the standard parabola $y^2 = 4ax$ is always at which point?

What are the coordinates of the focus of the parabola $y^2 = -16x$?

An equilateral triangle is inscribed in the parabola $y^2 = 4ax$ such that one of its vertices coincides with the vertex of the parabola. What is the length of each side of this triangle?

What is the length of the latus rectum for the parabola $y^2 = 2px$?

Find the vertex of the parabola $y^2 - 4y - 4x = 0$.

The locus of the center of a circle which touches the line $x + 2 = 0$ and the circle $(x - 2)^2 + y^2 = 4$ is a parabola. Find its latus rectum.

If a parabolic reflector is $20$ cm in diameter and $5$ cm deep, find the distance of the focus from the vertex.

Find the length of the focal chord of the parabola $y^2 = 4ax$ whose one end is $(am^2, 2am)$.

Find the eccentricity of the ellipse $\frac{x^2}{81} + \frac{y^2}{72} = 1$.

What are the coordinates of the foci for the ellipse $\frac{x^2}{144} + \frac{y^2}{169} = 1$?

For a vertical ellipse where the major axis is on the y-axis, what is the relation between $a, b,$ and $c$ (where $b > a$)?

Find the length of the minor axis of the ellipse $\frac{x^2}{5} + \frac{y^2}{9} = 1$.

If the length of the minor axis of an ellipse is equal to the distance between its foci, find the eccentricity.

Find the equation of the ellipse with the major axis on the $x$-axis, length 20, and eccentricity $e = \frac{1}{2}$.

What is the equation of the ellipse with ends of the major axis at $(\pm 3, 0)$ and ends of the minor axis at $(0, \pm 2)$?

An ellipse is drawn with its focus at $(1, 0)$, the corresponding directrix $x = 4$, and eccentricity $1/2$. Its equation is:

The area of the triangle inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the eccentric angles of whose vertices are $\alpha, \beta, \gamma$ is:

Find the coordinates of the foci of the ellipse $2x^2 + 3y^2 = 6$.

Find the foci of the hyperbola $\frac{y^2}{25} - \frac{x^2}{11} = 1$.

Find the eccentricity of the hyperbola $\frac{y^2}{1} - \frac{x^2}{3} = 1$.

What is the length of the conjugate axis of the hyperbola $\frac{y^2}{16} - \frac{x^2}{4} = 1$?

What is the vertex of the hyperbola $\frac{y^2}{4} - \frac{x^2}{9} = 1$ that lies on the positive y-axis?

A hyperbola has its conjugate axis of length $6$ and a focus at $(0, 5)$. What is its equation?

Determine the eccentricity of the hyperbola $16x^2 - y^2 = 16$.

Find the equation of the hyperbola with vertices $(\pm 2, 0)$ and foci $(\pm 4, 0)$.

Find the product of the distances from the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ to any point $(x_1, y_1)$ on it.

The eccentricity of the hyperbola $x^2 - 4y^2 = 1$ is:

The equation of the hyperbola with vertices at $(0, \pm 5)$ and directrices $y = \pm 25/7$ is:

What does the equation $x^2 - 2xy + y^2 = 0$ represent?

Is the equation $x^2 + y^2 + 1 = 0$ a real degenerate conic section?

The equation $(x+2)^2 + (y-5)^2 = 0$ represents which point?

What is the name of the section if $\alpha = \beta$ and the plane passes through the vertex?

The equation $x^2 + y^2 + 2x + 1 = 0$ represents which geometric figure?

If $ax^2 + 2hxy + by^2 = 0$ represents a single point (the origin), then which condition must hold for the discriminant $h^2 - ab$?

The equation $x^2 - 3x + 2 = 0$ represents which of the following in a 2D plane?

Find the area of the circumcircle of the triangle formed by the pair of lines $x^2 - 4xy + y^2 = 0$ and the line $x + y = 4\sqrt{6}$.

If the lines $x^2 - 2xy - y^2 = 0$ are rotated about the origin by $45^{\circ}$ in clockwise direction, the new equation represents:

Which of the following values of $a$ makes $ax^2 + 2xy + y^2 = 0$ a pair of imaginary lines?

Find the coordinates of the focus for $y^2 = 64x$.

Find the equation of the directrix for $y^2 = -4x$.

Find the focus of $x^2 = -40y$.

Find the length of the latus rectum for $x^2 = 100y$.

A chord of the parabola $y^{2} = 4ax$ is drawn perpendicular to its axis. If the length of this chord is $8a$, find the measure of the angle it subtends at the vertex of the parabola.

A chord of the parabola $y^{2} = 12x$ passes through the vertex and is inclined at an angle of $60^{\circ}$ to the axis of the parabola. What is the length of this chord?

Find the coordinates of the focus of a parabola whose latus rectum has endpoints $(2, 4)$ and $(2, -4)$.

The coordinates of the ends of a focal chord of $y^2 = 4ax$ are $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$. If $t_1 = 3$, find $t_2$.

The angle between the tangents at the ends of any focal chord of a parabola is:

What is the length of the latus rectum of a parabola whose focal chord segments are 3 and 6?

Calculate the length of the latus rectum of the parabola $y^2 = 64x$.

What is the length of the latus rectum of the parabola $x^2 = 15y$?

If the focus is at $(7, 0)$ and the vertex is at $(0, 0)$, what is the length of the latus rectum?

Find the length of the latus rectum of the parabola $y^2 = -10x$.

A parabola has its vertex at the origin and its axis along the $x$-axis. If the parabola passes through the point $(5, 5)$, determine the length of its latus rectum.

A parabola has its vertex at $(-2, 3)$ and its directrix is the line $y = \frac{21}{4}$. What is the length of its latus rectum?

The distance between the focus and the endpoint of the latus rectum for the parabola $y^2 = 4ax$ is:

Find the length of the latus rectum of the parabola $2x^2 + 3y = 0$.

The latus rectum of $y^2 = 4ax$ is a focal chord. If another focal chord $PQ$ has length $k$, what is the relationship between the semi-latus rectum $l$ and the segments $SP$ and $SQ$?

The length of the latus rectum of the parabola whose vertex is $(3, -2)$ and focus is $(3, 1)$ is:

In an ellipse, what is the value of $a^2 - c^2$?

If $a=b$ in an ellipse, the distance $c$ is zero. This specific conic is a:

Find $c$ if the equation of the ellipse is $9x^2 + 25y^2 = 225$.

If $c^2 = 49$ and $b^2 = 24$ for a hyperbola, find $a$.

If the distance between the foci of an ellipse is $14$ units and the length of the semi-major axis is $25$ units, find the length of the semi-minor axis.

Find the sum of the distances from any point on the ellipse to the two foci, given $a=15$ and $b=9$.

If $e = 0.5$ and $c = 2$, find the length of the semi-minor axis $b$.

The sum of the focal distances of any point on an ellipse is $20$ and the eccentricity is $0.5$. Find the distance from the center to a focus $c$.

If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$, find the length of its latus rectum.

In an ellipse, the focal distance of an end of the minor axis is $k$. What is the length of the semi-major axis $a$?

If the length of the major axis is 20 and the eccentricity is 0.5, what is the length of the minor axis?

Find the eccentricity of the ellipse $x^2/25 + y^2/9 = 1$.

If the eccentricity of an ellipse is $3/5$ and the semi-major axis is 10, find the semi-minor axis.

For an ellipse, if $b^2 = a^2(1 - e^2)$, then $e$ is the:

In an ellipse, the length of the minor axis is equal to the distance between a focus and the farther vertex. Find the eccentricity of the ellipse.

Calculate the eccentricity of the ellipse $4x^2 + 9y^2 = 1$.

Find the eccentricity of an ellipse whose major axis is 4 times the minor axis.

If the eccentricity of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $e$, and the distance between the foci is $d$, what is the length of the latus rectum in terms of $d$ and $e$?

If the focal distance of a point $P(x, y)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $r_1$ and $r_2$, and $r_1 r_2 = b^2$, then the point $P$ must be:

The eccentricity of the ellipse where the latus rectum is $1/n$ of the major axis is:

Find the length of the major axis for the ellipse $\frac{x^2}{100} + \frac{y^2}{400} = 1$.