Precalculus Conic Sections Test Solutions and Scoring (PDF)

Precalculus math test solutions: Conic Sections – Maths PDF

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This conics test answers for Precalculus covers conic sections (Common Core G-GPE.A.1, G-GPE.A.2, G-GPE.A.3), with a free PDF to print.

Worked solutions and scoring notes for the 9-question conic sections test. Use the scoring line under each solution to award partial credit.

Precalculus · Common Core State Standards: G-GPE.A.1, G-GPE.A.2, G-GPE.A.3

Conics test answers: worked solutions

Question 1 – Circles [2 pts]

(a) \((x+1)^{2}+(y-4)^{2}=9\). (b) \((x^{2}+8x+16)+(y^{2}-2y+1)=-8+16+1=9\): \((x+4)^{2}+(y-1)^{2}=9\).

(a) \((x+1)^{2}+(y-4)^{2}=9\); (b) center \((-4,1)\), radius \(3\)

Scoring: 1 point each.

Question 2 – Parabola parts [2 pts]

\(4p=-16\), \(p=-4\). The parabola opens down.

Focus \((0,-4)\); directrix \(y=4\)

Scoring: 1 point each.

Question 3 – Parabola from focus and directrix [2 pts]

Vertex \((0,0)\), \(p=-2\), opens left: \(y^{2}=4px\).

\(y^{2}=-8x\)

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

Question 4 – Ellipse foci [2 pts]

Divide by 400: \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\). \(c=\sqrt{25-16}=3\).

\(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\); foci \((\pm3,0)\)

Scoring: 1 point for standard form, 1 point for foci.

Question 5 – Read an ellipse [3 pts]

Foci at (±4, 0).

(a) \(a=6\), \(b=\sqrt{20}=2\sqrt5\). (b) \(c=\sqrt{36-20}=4\); foci \((\pm4,0)\). (c) \(2a=12\).

\(a=6,\ b=2\sqrt5\); \(c=4\), foci \((\pm4,0)\); \(12\)

Scoring: 1 point for each part.

Question 6 – Hyperbola parts [2 pts]

\(a=5\), \(b=12\). Vertices \((\pm5,0)\). Asymptotes \(y=\pm\frac{12}{5}x\).

\((\pm5,0)\); \(y=\pm\frac{12}{5}x\)

Scoring: 1 point each.

Question 7 – Hyperbola from vertices and foci [2 pts]

\(a=3\), \(c=5\), \(b^{2}=25-9=16\). The branches open up and down, so \(y^{2}\) comes first.

\(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\)

Scoring: 1 point for b squared, 1 point for the equation.

Question 8 – Satellite dish [3 pts]

p = 1.5 so the focus is 1.5 feet above the vertex.

(a) The rim point is \((3,1.5)\): \(9=4p(1.5)\), so \(p=1.5\) and \(x^{2}=6y\). (b) The focus is \((0,1.5)\). (c) \(y=\frac{4}{6}\approx0.67\).

\(x^{2}=6y\); \((0,1.5)\); \(0.67\text{ ft}\)

Scoring: 1 point for each part.

Question 9 – Classify conics [2 pts]

(a) Divide by 3: \(x^{2}+y^{2}=9\), a circle of radius 3. (b) Opposite signs: a hyperbola.

(a) circle; (b) hyperbola

Scoring: 1 point each.

Total score

Total

20 points

Key facts behind this conics test answers

Key facts

Circle: (x – h)^2 + (y – k)^2 = r^2.
Parabola: (x – h)^2 = 4p(y – k); focus is p from the vertex and the directrix is on the opposite side.
Ellipse: x^2/a^2 + y^2/b^2 = 1 with c^2 = a^2 – b^2; the foci lie on the major axis.
Hyperbola: x^2/a^2 – y^2/b^2 = 1 with c^2 = a^2 + b^2 and asymptotes y = plus or minus (b/a) x.
Complete the square to find the center or vertex.

How to use this conics test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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