Question

How many of the following statements are correct?

(a) The focus of $x^2=4ay$ is (0, a)

(b) The directrix of $y^2=-4ax$ is x + a = 0

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

(d) The parabolas $x^2=4ay$ and $y^2=4ax$ are equal.

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Answer 1

The parabolas $y^2=4ax,x^2=4ay,y^2=-4axand{x}^2=-4ay$ are the four standard parabolas.

If we know the details of $y^2=4ax$, we can derive the same for other three easily. We get $x^2=4ay$ by rotating $y^2=4ax$ through ${90}^{\circ }$ anti clockwise.

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

1) Focus of $x^2=4ay$ is (0, a)

$\Rightarrow$True

2)We want to find the directrix of $y^2=-4ax$

We get $y^2=-4ax$ by rotating $x^2=4ay$ through ${90}^{\circ }or{y}^2=4axby{180}^{\circ }$

It is the reflection of $y^2=4ax$ about y-axis. so the directrix become x = a from x = -a

$\Rightarrow$ 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 $x^2=-4ay$

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

$\Rightarrow$ y = -a

(x, y) lies on $x^2=-4ay$

$\Rightarrow x^2=-4ax-a=4a^2$

$\Rightarrow x=\pm 2a$

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

$\Rightarrow$ statement 3 is false

4) We say two parabolas are equal if they have same latusrectam. $y^2=4axand{x}^2=4ay$ have the same latus rectum (4a)

$\Rightarrow$ statement 2 is wrong.

So the statements 1 and 3 are correct.