EXAMPLE 3 Write an equation of a translated parabola

Write an equation of the parabola whose vertex is at (–2, 3) and whose focus is at (–4, 3).

SOLUTION

STEP 1

Determine the form of the equation. Begin by making a rough sketch of the parabola. Because the focus is to the left of the vertex, the parabola opens to the left, and its equation has the form (y – k)2 = 4p(x – h) where p < 0.

EXAMPLE 3 Write an equation of a translated parabola

STEP 2

Identify h and k. The vertex is at (–2, 3), so h = –2 and k = 3.

STEP 3

Find p. The vertex (–2, 3) and focus (–4, 3) both lie on the line y = 3, so the distance between them is | p | = | –4 – (–2) | = 2, and thus p = +2. Because p < 0, it follows that p = –2, so 4p = –8.

EXAMPLE 3 Write an equation of a translated parabola

The standard form of the equation is (y – 3)2 = –8(x + 2).

ANSWER

EXAMPLE 4 Write an equation of a translated ellipse

Write an equation of the ellipse with foci at (1, 2) and (7, 2) and co-vertices at (4, 0) and (4, 4).

SOLUTION

STEP 1

Determine the form of the equation. First sketch the ellipse. The foci lie on the major axis, so the axis is horizontal. The equation has this form:

(x – h)2

a2 +(y – k)2

b2 = 1

EXAMPLE 4 Write an equation of a translated ellipse

STEP 2

Identify h and k by finding the center, which is halfway between the foci (or the co-vertices)

(h, k) = 1 + 7 2 + 22 2 )( , = (4, 2)

STEP 3Find b, the distance between a co-vertex and the center (4, 2), and c, the distance between a focus and the center. Choose the co-vertex (4, 4) and the focus (1, 2): b = | 4 – 2 | = 2 and c = | 1 – 4 | = 3.

EXAMPLE 4 Write an equation of a translated ellipse

STEP 4

Find a. For an ellipse, a2 = b2 + c2 = 22 + 32 = 13, so a =

13

ANSWER

The standard form of the equation is

(x – 4)2

13 +(y – 2)2

4 = 1

EXAMPLE 5 Identify symmetries of conic sections

Identify the line(s) of symmetry for each conic section in Examples 1 – 4.

SOLUTION

For the circle in Example 1, any line through the center (2, –3) is a line of symmetry.

For the hyperbola in Example 2, x = –1 and y = 3 are lines of symmetry

EXAMPLE 5 Identify symmetries of conic sections

For the parabola in Example 3, y = 3 is a line of symmetry.

For the ellipse in Example 4, x = 4 and y = 2 are lines of symmetry.

GUIDED PRACTICE for Examples 3, 4 and 5

Write the equation of parabola with vertex at (3, – 1) and focus at (3, 2).

5.

The standard form of the equation is (x – 3)2 = 12(y + 1).

ANSWER

Write the equation of the hyperbola with vertices at (–7,3) and (–1, 3) and foci at (–9, 3) and (1, 3).

6.

ANSWER

The standard form of the equation is

(x + 4)2

9 –(y – 3)2

16 = 1

GUIDED PRACTICE for Examples 3, 4 and 5

7. (x – 5)2

64 + (y)2

16 = 1

Identify the line(s) of symmetry for the conic section.

x = 5 and y = 0.

ANSWER

GUIDED PRACTICE for Examples 3, 4 and 5

8. (x + 5)2 = 8(y – 2).

x = –5

ANSWER

Identify the line(s) of symmetry for the conic section.

GUIDED PRACTICE for Examples 3, 4 and 5

9. (x – 1)2

49 –(y – 2)2

121 = 1

x = 1 and y = 2.

ANSWER

Identify the line(s) of symmetry for the conic section.