Precalculus Conic Sections Worksheet (PDF)

Precalculus math worksheet: Conic Sections – Maths PDF

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This conics practice worksheet 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.

These 20 problems practice conic sections (Common Core G-GPE.A.1, G-GPE.A.2 and G-GPE.A.3). You will write equations of circles, parabolas, ellipses and hyperbolas, find foci, directrices and asymptotes, complete the square and solve applications.

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

Conics practice worksheet: exercises 1 to 10

Exercise 1 – Equation of a circle

A circle on the plane.

Write the equation of the circle with center \((3,-2)\) and radius \(5\).

Exercise 2 – Center and radius

Find the center and radius of \(x^{2}+y^{2}-6x+4y-12=0\).

Exercise 3 – Circle through a point

A circle has center \((-2,6)\) and passes through \((1,2)\). Write its equation.

Exercise 4 – Parts of a parabola

Parabola x² = 8y.

Find the vertex, focus and directrix of \(x^{2}=8y\).

Exercise 5 – Parabola opening left

Find the focus and directrix of \(y^{2}=-12x\).

Exercise 6 – Equation from focus and directrix

Write the equation of the parabola with focus \((0,-5)\) and directrix \(y=5\).

Exercise 7 – Shifted parabola

Find the vertex, focus and directrix of \((x-2)^{2}=4(y+1)\).

Exercise 8 – Complete the square for a parabola

Write \(y=x^{2}-4x+7\) in the form \((x-h)^{2}=4p(y-k)\) and find the focus.

Exercise 9 – Parts of an ellipse

For \(\frac{x^{2}}{100}+\frac{y^{2}}{36}=1\), find \(a\), \(b\), \(c\), the vertices and the foci.

Exercise 10 – Vertical ellipse

Find the foci of \(\frac{x^{2}}{20}+\frac{y^{2}}{36}=1\).

Exercises 11 and up

Exercise 11 – Ellipse from vertices and foci

Write the equation of the ellipse with vertices \((\pm6,0)\) and foci \((\pm4,0)\).

Exercise 12 – Shifted ellipse

For \(\frac{(x-1)^{2}}{9}+\frac{(y+2)^{2}}{4}=1\), find the center and the foci.

Exercise 13 – Standard form of an ellipse

Write \(4x^{2}+9y^{2}=36\) in standard form.

Exercise 14 – Parts of a hyperbola

For \(\frac{x^{2}}{49}-\frac{y^{2}}{576}=1\), find the vertices, the foci and the asymptotes.

Exercise 15 – Vertical hyperbola

For \(\frac{y^{2}}{25}-\frac{x^{2}}{144}=1\), find the foci and the asymptotes.

Exercise 16 – Hyperbola from vertices and foci

Write the equation of the hyperbola with vertices \((\pm4,0)\) and foci \((\pm5,0)\).

Exercise 17 – Identify the conic

Identify each as a circle, parabola, ellipse or hyperbola: (a) \(x^{2}+y^{2}=9\); (b) \(y=2x^{2}\); (c) \(4x^{2}-y^{2}=16\); (d) \(5x^{2}+2y^{2}=10\).

Exercise 18 – Satellite dish

Cross-section of a dish.

A parabolic satellite dish is \(4\) ft across and \(1\) ft deep. Model it by \(x^{2}=4py\) with the vertex at the origin. Find \(p\) and how far above the vertex the receiver (the focus) should be.

Exercise 19 – Whispering gallery

A whispering gallery with two foci.

A whispering gallery is a half-ellipse hall \(100\) ft long and \(40\) ft wide. Whispers at one focus carry to the other. Find the distance between the foci to the nearest tenth of a foot.

Exercise 20 – Elliptical arch

A semi-elliptical arch.

A bridge arch is the upper half of an ellipse \(60\) ft wide and \(20\) ft high. Write its equation with the center at the origin, and find the height \(15\) ft from the center to the nearest hundredth.

Key facts behind this conics practice worksheet

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 practice worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the conics practice worksheet as a PDF with the button below and keep it with your notes.

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