Chapter 9: Q. 27 (page 772)

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

Short Answer

Expert verified

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

Step by step solution

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,

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Probability and Statistics

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.directrixy=y0, focusx1,y1, wherey0≠y1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Probability and Statistics

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

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