Practice Questions

Analyze the equation $x^2 = -10y$ to find the coordinates of the focus of the parabola.

Determine the equation of the directrix for the parabola $y^2 = 14x$.

Propose an argument to explain why the directrix of a parabola can never pass through its focus.

Analyze the equation $\frac{x^2}{49} - \frac{y^2}{25} = 1$ to find the coordinates of the vertices of the hyperbola.

Define a circle.

Name the conic section obtained when a plane intersects a double-napped cone at an angle $\beta = 90^{\circ}$ with the vertical axis of the cone.

State the standard equation of a parabola with its vertex at the origin and its axis of symmetry along the positive x-axis.

Identify the center and radius of the circle given by the equation $(x - 3)^2 + (y + 5)^2 = 49$.

Calculate the radius of the circle given by the equation $(x - 5)^2 + (y + 3)^2 = 49$.

Explain the conditions, using the angles $\alpha$ and $\beta$, under which the intersection of a plane with a double-napped cone results in (a) an ellipse, (b) a parabola, and (c) a hyperbola. (Here $\alpha$ is the angle of the generator with the axis and $\beta$ is the angle of the plane with the axis).

Calculate the eccentricity of the ellipse given by the equation $25x^2 + 9y^2 = 225$.

Justify why the eccentricity $e$ of any parabola is always equal to 1, based on its definition.

Identify the coordinates of the vertices and the foci for the ellipse given by the equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$.

Create a mathematical model for a parabolic reflector. The reflector is designed such that it is 24 cm in diameter and 9 cm deep. Formulate the equation of the parabola and determine the position of its focus.

Design a plan for an archway in the shape of a semi-ellipse. The base of the arch must be 10 meters wide. A tall vehicle, 2.5 meters high, needs to pass through the arch. To ensure safe passage, the vehicle must be able to drive through at a distance of 1 meter from the outer edge of the arch. Formulate the equation for the arch and calculate the minimum required height at its center.

Prove that for any ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (with $a>b$), the length of the latus rectum is $\frac{2b^2}{a}$. Your proof must be derived from the definition and equation of the ellipse.

Summarize the key properties of a parabola defined by the equation $x^2 = -4ay$, where $a > 0$. Include its vertex, focus, equation of the directrix, axis of symmetry, and the direction it opens.

Find the equation of a circle which is concentric with the circle $x^2 + y^2 - 8x + 12y + 15 = 0$ and has an area of $25\pi$ square units.

Define the eccentricity of a hyperbola.

For the parabola given by the equation $y^2 = 16x$, list the coordinates of the focus, the equation of the directrix, and the length of the latus rectum.

Determine the equation of the ellipse whose foci are at $(0, \pm 6)$ and the length of the minor axis is 16.

Describe the standard form of the equation of an ellipse with its center at the origin and major axis along the y-axis. List the formulas for its foci, vertices, eccentricity, and the length of its latus rectum in terms of $a$ and $b$.

Calculate the equation of the parabola with its vertex at the origin, the axis of symmetry along the y-axis, and passing through the point $(-4, -8)$.

Describe the major axis and minor axis of an ellipse. State which axis contains the foci.

What is the relationship between the semi-major axis $a$, the semi-minor axis $b$, and the distance $c$ from the center to a focus for an ellipse?

Solve for the center and radius of the circle defined by the equation $3x^2 + 3y^2 + 6x - 12y - 1 = 0$.

Critique the statement: "A circle is a special case of an ellipse." Justify your answer using the standard equation of an ellipse.

Evaluate why a hyperbola has two separate branches while an ellipse is a single closed curve, by considering their respective geometric definitions.

Find the equation of the hyperbola with foci at $(\pm 10, 0)$ and an eccentricity of $e = \frac{5}{3}$.

For the hyperbola $16y^2 - 9x^2 = 144$, analyze the equation to find the coordinates of the foci, the vertices, the eccentricity, and the length of the latus rectum.

Formulate the equation of a circle passing through the origin $(0,0)$ and making intercepts $p$ and $q$ on the x-axis and y-axis respectively. Justify your result.

Derive the equation of a parabola with its vertex at $(h, k)$ and its axis of symmetry parallel to the x-axis, opening to the right.

A rod of length 12 cm moves with its ends always touching the coordinate axes. A point P on the rod is 4 cm from the end in contact with the x-axis. Derive the equation of the locus of P and identify the conic section.

An archway is in the shape of a semi-ellipse. The base of the arch is 10 m wide, and the highest point of the arch is 4 m above the ground. Calculate the height of the arch at a point 2 m from the center of the base.

A comet has a hyperbolic orbit with the Sun at one focus. The equation of its path is $\frac{x^2}{144} - \frac{y^2}{400} = 1$, where distances are in millions of kilometers. When the comet is at one of the vertices of its path, how far is it from the Sun? Formulate a general principle for the closest distance of an object in a hyperbolic orbit to the star at its focus.

For the hyperbola $\frac{y^2}{9} - \frac{x^2}{16} = 1$, identify the orientation of the transverse axis and state the length of the latus rectum.

An equilateral triangle is inscribed in the parabola $y^2 = 8ax$, with one of its vertices at the vertex of the parabola. Calculate the length of a side of the triangle.

Formulate the condition for the line $y = mx + c$ to be tangent to the circle $x^2 + y^2 = r^2$.

Explain the difference between the transverse axis and the conjugate axis of a hyperbola.

Explain the concept of 'degenerate conic sections'. Describe the three types of degenerate conics and the conditions under which they are formed when a plane intersects the vertex of a cone.

A parabolic arch has a height of 25 m and a base width of 40 m. Calculate the height of the arch at a horizontal distance of 10 m from the center of the base.

Calculate the equation of the circle that passes through the points $(1, 2)$ and $(3, 4)$ and whose center lies on the line $3x + y = 11$.

A point $P(x, y)$ moves such that the sum of its distances from two fixed points $F_1(0, -c)$ and $F_2(0, c)$ is a constant $2a$, where $2a > 2c$. Derive the equation of the locus of $P$ and identify the conic.

Justify the conditions on the angles $\alpha$ and $\beta$ that result in the three non-degenerated conic sections (ellipse, parabola, hyperbola), where $\alpha$ is the semi-vertical angle of the cone and $\beta$ is the angle the cutting plane makes with the cone's axis.

An equilateral triangle is inscribed in the parabola $y^2 = 4ax$ such that one of its vertices is at the vertex of the parabola. Derive the length of a side of this triangle.