Parabola Focus Directrix and Tangent Lines Test Answer Key with Scoring – Geometry

Question 1 Equation from the Focus (2 points)

The vertex is (0, 0) and p = 10, so x² = 40y.

x² = 40y

Scoring: 1 point for p, 1 point for the equation.

Question 2 Parabola Opening Left or Right (3 points)

The vertex is halfway: (−5, 2). p = −3 − (−5) = 2, so the parabola opens right.

(y − 2)² = 4(2)(x + 5).

(y − 2)² = 8(x + 5)

Scoring: 1 point for the vertex, 1 point for p, 1 point for the equation.

Question 3 Complete the Square (3 points)

x² + 10x = −12y − 49. Add 25: (x + 5)² = −12y − 24 = −12(y + 2).

Vertex (−5, −2), p = −3. Focus (−5, −5). Directrix y = 1.

(x + 5)² = −12(y + 2); focus (−5, −5); directrix y = 1

Scoring: 1 point for completing the square, 1 point for the focus, 1 point for the directrix.

Question 4 Is the Point on the Parabola? (2 points)

Distance from P to the focus (0, 5): 10. Distance from P to the directrix y = −5: 5 + 5 = 10.

Yes, P is on the parabola.

Scoring: 1 point for each distance.

Question 5 Tangent Segment Length (2 points)

OT ⊥ PT, so PT = √(37² − 12²) = √1225 = 35.

35

Scoring: 1 point for the right triangle, 1 point for the answer.

Question 6 Tangent Segments with Algebra (3 points)

4x − 9 = 2x + 11, so 2x = 20 and x = 10. PA = 31.

OP = √(31² + 12²) = √1105 ≈ 33.24.

x = 10; PA = 31; OP ≈ 33.2

Scoring: 1 point for x, 1 point for PA, 1 point for OP.

Question 7 Telescope Dish (3 points)

The edge point is (18, 6). x² = 4py gives 324 = 4p(6) = 24p, so p = 13.5.

13.5 inches above the vertex

Scoring: 1 point for the edge point, 1 point for the equation, 1 point for p.

Question 8 Why the Construction Works (2 points)

∠OTP is inscribed in a semicircle of the circle with diameter OP, so it is a right angle. A line perpendicular to a radius at its endpoint on the circle is tangent.

∠OTP = 90°, so PT ⊥ OT and PT is tangent.

Scoring: 1 point for the semicircle angle, 1 point for the perpendicular-radius conclusion.