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Q. 22

Expert-verified

Calculus

Found in: Page 772

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix x = 3, focus (0, 1)$

The equation is $(y - 1)^{2} = - 6 x + 9 .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

We are given,

$directrix x = 3, focus (0, 1)$

Step 2. Distance formula.

Now,

$Let (x, y) be any point on the parabola.$

We know,

$Formula for the distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} Distance = \sqrt{(0 - x)^{2} + (1 - y)^{2}} [since x_{1} = x, y_{1} = y, x_{2} = 0, y_{2} = 1]$

Now, the distance between the point and directrix $|x - 3|$

Then,

$\sqrt{(0 - x)^{2} + (1 - y)^{2}} = | x - 3 |$

Step 3. Final answer.

On simplifying the equation,

${(\sqrt{(0 - x)^{2} + (1 - y)^{2}})}^{2} = (| x - 3 |)^{2} (0 - x)^{2} + (1 - y)^{2} = (x - 3)^{2} x^{2} + 1 + y^{2} - 2 y = (x^{2} - 6 x + 9) 1 + y^{2} - 2 y = - 6 x + 9 y^{2} - 2 y + 1 = - 6 x + 9 (y - 1)^{2} = - 6 x + 9$

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