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Chapter 9: Q. 23 (page 772)

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22鈥�31.

directrixy=0,focus(0,1)

Short Answer

Expert verified

The equation isy=12x2+12.

Step by step solution

01

Step 1. Given information.

We are given,

directrixy=0, focus(0,1)

02

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=x2-x12+y2-y12Distance=(0-x)2+(1-y)2sincex1=x,y1=y,x2=0,y2=1

Now the distance between the point and the directrix isy-0.

Therefore,

(0-x)2+(1-y)2=|y-0|

03

Step 3. Final answer.

On simplifying the equation,

(0-x)3+(1-y)32=(|y|)2(0-x)2+(1-y)2=y2x2+1-2y+y2=y2x2+1-2y+2y=0+2yx2+1=2y2y2=x2+12y=12x2+12

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