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

Geometry Parabola Tangents Test: questions 1–5

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.

Geometry Parabola Tangents Test: questions 6–9

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)

How to use this geometry parabola tangents test

Take this geometry parabola tangents test without notes, then grade it with the test answer key out of 20 points.

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