Chapter 9: Q. 23 (page 772)

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

Short Answer

Expert verified

The equation is $y = \frac{1}{2} x^{2} + \frac{1}{2}$ .

Step by step solution

Step 1. Given information.

We are given,

$directrix y = 0, focus (0, 1)$

Step 2. Distance Formula.

Now,

$Let (x, y) be any point on the parabola. Now find the distance between the point (x, y) and the focus (0, 1) . Formula for the distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} Distance = \sqrt{(0 - x)^{2} + (1 - y)^{2}} [since x_{1} = x, y_{1} = y, x_{2} = 0, y_{2} = 1]$

Now the distance between the point and the directrix is $|y - 0|$ .

Therefore,

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

Step 3. Final answer.

On simplifying the equation,

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

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

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.directrixy=0,focus(0,1)

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

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Applied Mathematics

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.
$directrix y = 0, focus (0, 1)$