# To show that the for a parabola, x 2 = − 4 p y x 2 = − 4 p y

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter B, Problem 47E
To determine

## To calculate: To show that the for a parabola, x2=−4py  x2=−4py

Expert Solution

It is proved that x2=4py

### Explanation of Solution

Given information: P(x,y) is any point on the parabola with focus (0,p) and directrix y=p

Formula Used:

Parabola is a set of points in the plane that are equidistant from a fixed point F called the focus and a fixed line called the directrix

Calculation:

Consider parabola with focus F(0,p) and directrix y=p

Let P(x,y) is any point on the parabola

Distance between the point and focus is FP

FP=(x0)2+(yp)2FP=x2+y2+p22yp

Let us assume that thedistance between the point and directrix y=p is K

K=y+p02+12K=y+p12K=y+p

Now, the definition of parabola,

FP=K

Thus,

x2+y2+p22py=y+p(x2+y2+p22py)2=(y+p)2x2+y2+p22py=y2+p2+2pyx22py=2pyx2=4py

Conclusion:

Hence, proved that x2=4py

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