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Q38.

Expert-verified

Algebra 2

Found in: Page 424

Book edition Middle English Edition

Author(s) Carter

Pages 804 pages

ISBN 9780079039903

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Short Answer

Write an equation for the parabola having focus $(- 4, - 2)$ and directrix $x = - 8$ . Then draw the graph.

The required equation of the parabola is $x = \frac{1}{8} {(y + 2)}^{2} - 6$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Write down the given information.

The given parabola has focus $(- 4, - 2)$ and directrix $x = - 8$ .

Step 2. Concept used.

If $P (x, y)$ be any point on parabola having focus $(f_{1}, f_{2})$ and directrix $x = a$ then:

Distance of point $P (x, y)$ from focus $(f_{1}, f_{2}) =$ Distance of point $P (x, y)$ from $(a, y)$

Step 3. Calculation.

Since the given parabola has focus $(- 4, - 2)$ and directrix $x = - 8$ . Therefore, apply the concept stated above,

Distance of point $P (x, y)$ from focus $(- 4, - 2)$ Distance of point $P (x, y)$ from $(- 8, y)$

$\begin{matrix} \sqrt{{(x + 4)}^{2} + {(y + 2)}^{2}} = \sqrt{{(x + 8)}^{2} + {(y - y)}^{2}} \\ {(x + 4)}^{2} + {(y + 2)}^{2} = {(x + 8)}^{2} + {(y - y)}^{2} .... (Squaring) \\ {(x + 4)}^{2} + {(y + 2)}^{2} = {(x + 8)}^{2} \\ {(y + 2)}^{2} = {(x + 8)}^{2} - {(x + 4)}^{2} \\ {(y + 2)}^{2} = (2 x + 12) (4) \\ {(y + 2)}^{2} = 8 x + 48 \\ x = \frac{1}{8} {(y + 2)}^{2} - 6 \end{matrix}$

Hence, $x = \frac{1}{8} {(y + 2)}^{2} - 6$ is the required equation of the parabola.

Step 4. Sketch the graph of the parabola.

The graph of the parabola $x = \frac{1}{8} {(y + 2)}^{2} - 6$ is shown below.

Step 5. Conclusion.

The required equation of the parabola is $x = \frac{1}{8} {(y + 2)}^{2} - 6$ .

Most popular questions for Math Textbooks

COMMUNICATION A microphone is placed at the focus of a parabolic reflector to collect sound for the television broadcast of a World Cup soccer game. Write an equation for the cross section, assuming that the focus is at the origin and the parabola opens to the right.

Find the distance between the pair of points $(0.5, 1.4) and (1.1, 2.9)$ with the given coordinates.

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation ${(y - 8)}^{2} = - 4 (x - 4)$ . Then find the length of latus rectum and graph the parabola.

Use the equation $x = 3 y^{2} + 4 y + 1$ to find the axis of symmetry.

Use the equation $x = 3 y^{2} + 4 y + 1$ to find the y-intercept.

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