Parabolas from Focus and Directrix, and Tangent Lines to a Circle: Geometry Math Test – PDF Download

Question 1: Equation from a focus and a directrix (2 points)

A parabola has focus \(F(0,5)\) and directrix \(y=-5\), as shown.

1. Write the equation of the parabola. (1 point)
2. Show that \(P(10,5)\) is on the parabola. (1 point)

Question 2: Focus and directrix from an equation (2 points)

Find the focus and the directrix of the parabola \(x^{2}=24y\). (1 point each)

Question 3: A sideways parabola (2 points)

Write the equation of the parabola with focus \((-3,0)\) and directrix \(x=3\).

Question 4: Parabola with a shifted vertex (3 points)

A parabola has focus \((1,7)\) and directrix \(y=3\).

1. Find the vertex and p, and write the equation. (2 points)
2. Find the y-coordinate of the point on the parabola with x-coordinate 5. (1 point)

Question 5: Length of a tangent (2 points)

In the figure, \(\overline{PT}\) is tangent to the circle with center O. The radius is 9 and \(OP=15\). Find PT.

Question 6: Two tangent segments (2 points)

Segments \(\overline{PA}\) and \(\overline{PB}\) are tangent to the circle at A and B, with \(PA=2x+7\) and \(PB=4x-5\). Find x and PA.

Question 7: Constructing a tangent (3 points)

Point P is outside the circle with center O.

1. Describe a compass-and-straightedge construction of the tangent lines from P to the circle. (2 points)
2. Explain why each line you construct is tangent. (1 point)

Question 8: Angles formed by two tangents (2 points)

Tangents \(\overline{PA}\) and \(\overline{PB}\) touch the circle with center O at A and B, and \(m\angle APB=48^{\circ}\). Find (a) \(m\angle AOB\) and (b) \(m\angle OPA\). (1 point each)

Question 9: A satellite dish (2 points)

A dish has a parabolic cross-section with its vertex at the origin, opening upward. The receiver is at the focus, 2 ft above the vertex. (a) Write the equation of the cross-section. (1 point) (b) The dish is 10 ft wide. How deep is it? (1 point)

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