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Chapter 9: Q. 24 (page 772)

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

directrixx=-1,focus(2,5)

Short Answer

Expert verified

The equation is(y-5)2=6x-3.

Step by step solution

01

Step 1. Given information.

It is given,

directrixx=-1,focus(2,5)

02

Step 2. Distance formula.

Now,

Let(x,y)be any point on the parabola.Formula for the distance=x2-x12+y2-y12

The distance between the point and the focus is,

Distance=(2-x)2+(5-y)2sincex1=x,y1=y,x2=2,y2=5

Now the distance between the point and directrix is |x+1|[sincex=-1]

Therefore,

(2-x)2+(5-y)2=|x+1|

03

Step 3. Final answer.

On simplifying the equation,

(2-x)2+(5-y)22=(|x+1|)2(2-x)2+(5-y)2=(x+1)24-4x+x2+25-10y+y2=x2+2x+1y2-10y+29-4x=2x+1y2-10y+29-4x+4x=6x+1y2-10y+29=6x+1(y-5)2+4=6x+1(y-5)2+4-4=6x+1-4(y-5)2=6x-3

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