Chapter 9: Q. 26 (page 772)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

The given values are:

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

Step 2. Distance formula.

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

Co-ordinates of the focus is $(x_{1}, y_{1})$

The distance between the point and the focus is,

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

Step 3. Final answer.

On simplifying the equation,

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Distance formula.

Step 3. Final answer.

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Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.directrixx=x0, focusx1,y1,wherex0≠x1.

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

Statistics

Calculus

Discrete Mathematics

Decision Maths

Pure Maths

Geometry

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