Q. 27 Use Cartesian coordinates to exp... [FREE SOLUTION]

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Chapter 9: Q. 27 (page 772)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

The given values are,

$directrix y = y_{0}, focus (x_{1}, y_{1}), where y_{0} 鈮� y_{1}$

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,

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

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

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91影视

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

Geometry

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Logic and Functions

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鈥�31.
$directrix y = y_{0}, focus (x_{1}, y_{1}), where y_{0} 鈮� y_{1}$