Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix y = y_{0}, focus (x_{1}, y_{1}), where y_{0} \neq y_{1}$

The equation is $y = \frac{1}{2} \cdot \frac{{(x - x_{1})}^{2}}{(y_{1} - y_{0})} + \frac{(y_{0} + y_{1})}{2}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given values are,

$directrix y = y_{0}, focus (x_{1}, y_{1}), where y_{0} \neq y_{1}$

Step 2. Distance formula.

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

Let $(x_{1}, y_{1})$ be the focus.

Therefore, by distance formula,

$Distance = \sqrt{{(x_{1} - x)}^{2} + {(y_{1} - y)}^{2}} [since x_{1} = x, y_{1} = y, x_{2} = x_{1}, y_{2} = y_{1}] Now the distance between the point and the directrix is |y - y_{0}| . T h e r e f o r e, \sqrt{{(x_{1} - x)}^{2} + {(y_{1} - y)}^{2}} = |y - y_{0}|$

Step 3. Final answer.

On simplifying the equation,

${(\sqrt{{(x_{1} - x)}^{3} + {(y_{1} - y)}^{3}})}^{2} = {(|y - y_{0}|)}^{2} {(x - x_{1})}^{2} + {(y - y_{1})}^{2} = {(y - y_{0})}^{2} {(x - x_{1})}^{2} = (y^{2} + y_{0}^{2} - 2 y y_{0}) - (y^{2} + y_{1}^{2} - 2 y y_{1}) {(x - x_{1})}^{2} = (y_{0}^{2} - y_{1}^{2}) - 2 y (y_{0} - y_{1}) \frac{{(x - x_{1})}^{2}}{(y_{0} - y_{1})} = \frac{(y_{0} - y_{1}) ((y_{0} + y_{1}) - 2 y)}{(y_{0} - y_{1})} \frac{{(x - x_{1})}^{2}}{(y_{0} - y_{1})} = (y_{0} + y_{1}) - 2 y o r - 2 y = \frac{{(x - x_{1})}^{2}}{(y_{0} - y_{1})} - (y_{0} + y_{1}) \frac{- 2 y}{- 2} = \frac{1}{- 2} \cdot \frac{{(x - x_{1})}^{2}}{(y_{0} - y_{1})} - \frac{1}{- 2} \cdot (y_{0} + y_{1}) y = \frac{1}{2} \cdot \frac{{(x - x_{1})}^{2}}{(y_{1} - y_{0})} + \frac{(y_{0} + y_{1})}{2}$

Find Study Materials for

Create Study Materials

Select your language

Calculus

Answers without the blur.

Short Answer

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix y = y_{0}, focus (x_{1}, y_{1}), where y_{0} \neq y_{1}$

Step by Step Solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Calculus

Answers without the blur.

Short Answer

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31. directrix y=y0, focus x1,y1, where y0≠y1

Step by Step Solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$directrix y = y_{0}, focus (x_{1}, y_{1}), where y_{0} \neq y_{1}$