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Question

How many of the following statements are correct?

(a) The focus of x2=4ay is (0, a)

(b) The directrix of y2=4ax is x + a = 0

(c) The end points of latus rectum of x2=4ay is (a, 2a) and (a, -2a)

(d) The parabolas x2=4ay and y2=4ax are equal.


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Solution

The parabolas y2=4ax,x2=4ay,y2=4ax and x2=4ay are the four standard parabolas.

If we know the details of y2=4ax, we can derive the same for other three easily. We get x2=4ay by rotating y2=4ax through 90 anti clockwise.

The coordinates of focus becomes (0, a) from (a, 0). Directrix become y + a = 0 from x + a = 0.

1) Focus of x2=4ay is (0, a)

True

2)We want to find the directrix of y2=4ax

We get y2=4ax by rotating x2=4ay through 90 or y2=4ax by 180

It is the reflection of y2=4ax about y-axis. so the directrix become x = a from x = -a

not correct

Lat us rectum is the chord passing through focus and perpendicular to axis. Now, the end points of latus rectum is the points where latus rectum meet the parabola. Let us draw it for x2=4ay

.

Let p(x, y) be one of the end points we can see that the y-coordinate of P and focus are same.

y = -a

(x, y) lies on x2=4ay

x2=4axa=4a2

x=±2a

So the end points are (2a, -a) and (-2a, -a)

statement 3 is false

4) We say two parabolas are equal if they have same latusrectam. y2=4ax and x2=4ay have the same latus rectum (4a)

statement 2 is wrong.

So the statements 1 and 3 are correct.


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