Conic Sections Test Answer Key with Scoring – Precalculus

Question 1 Circle in General Form (2 points)

(x² + 10x + 25) + (y² − 6y + 9) = −18 + 25 + 9, so (x + 5)² + (y − 3)² = 16.

Answer: center (−5, 3), radius 4

Scoring: 1 point for completing both squares, 1 point for the center and radius.

Question 2 A Downward Parabola (2 points)

The directrix is 3 units above the vertex, so the parabola opens downward with p = −3. The equation is (x + 1)² = 4(−3)(y − 4) = −12(y − 4). The focus is (−1, 4 − 3) = (−1, 1).

Answer: (x + 1)² = −12(y − 4); focus (−1, 1)

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

Question 3 Reading an Ellipse (3 points)

The larger denominator 36 is under y, so the major axis is vertical with a = 6 and b = √20 = 2√5. Then c² = 36 − 20 = 16 and c = 4.

(a) Vertices (0, ±6). (b) Minor axis endpoints (±2√5, 0). (c) Foci (0, ±4).

Answer: (0, ±6); (±2√5, 0); (0, ±4)

Scoring: 1 point for each part.

Question 4 A Hyperbola from Foci (3 points)

(a) c = 7 and a = 4, so b² = c² − a² = 49 − 16 = 33. The equation is x²16 − y²33 = 1.

(b) y = ±bax = ±√334x.

Answer: x²16 − y²33 = 1; asymptotes y = ±√334x

Scoring: 1 point for b², 1 point for the equation, 1 point for the asymptotes.

Question 5 Identify the Conic (2 points)

4(x² − 6x) + (y² + 4y) = −24. Complete the squares: 4(x − 3)² + (y + 2)² = −24 + 36 + 4 = 16, so (x − 3)²4 + (y + 2)²16 = 1.

It is an ellipse with center (3, −2), vertical major axis, a = 4, b = 2 and c = √12 = 2√3. The foci are (3, −2 ± 2√3).

Answer: ellipse; center (3, −2); foci (3, −2 ± 2√3)

Scoring: 1 point for the standard form and the type, 1 point for the center and foci.

Question 6 Parabolic Mirror (2 points)

The rim point is (12, 4). In x² = 4py: 144 = 16p, so p = 9. The equation is x² = 36y and the focus is at (0, 9), so the source sits 9 centimeters from the vertex.

Answer: x² = 36y; 9 cm from the vertex

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

Question 7 Running Track (3 points)

(a) a = 60 and b = 40: x²3600 + y²1600 = 1.

(b) c² = 3600 − 1600 = 2000, so c = 20√5 ≈ 44.7 m.

(c) The foci are 2c ≈ 89.4 m apart.

Answer: x²3600 + y²1600 = 1; about 44.7 m; about 89.4 m

Scoring: 1 point for each part.

Question 8 Checking the Definition (3 points)

(a) c² = 49 − 40 = 9, so the foci are (±3, 0).

(b) 049 + (2√10)²40 = 4040 = 1.

(c) The distance from (0, 2√10) to (3, 0) is √(9 + 40) = 7, and to (−3, 0) it is also 7. The sum is 14 = 2(7) = 2a.

Answer: foci (±3, 0); each focal distance is 7, with sum 14

Scoring: 1 point for each part.