Let P ( x 1 , y 1 ) and Q ( x 2 , y 2 ) , where y 1 0 , y 2 0 , be the end points of the latus rectum of the ellipse x 2 + 4 y 2 = 4 . The equations of parabolas with latus rectum P Q are [2008]

Question

Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ , where $y_{1} < 0, y_{2} < 0$ , be the end points of the latus rectum of the ellipse $x^{2} + 4 y^{2} = 4 .$ The equations of parabolas with latus rectum $P Q$ are [2008]

Smart Achievers · Accepted Answer

(2, 3)

$Given ellipse is x^{2} + 4 y^{2} = 4$

$or \frac{x^{2}}{2^{2}} + \frac{y^{2}}{1} = 1 \Rightarrow a = 2, b = 1$

$∴$ $e = \sqrt{1 - \frac{1}{4}} = \frac{\sqrt{3}}{2}$ $∴$ $a e = \sqrt{3}$

$As per question P \equiv (a e, - \frac{b^{2}}{a}) = (\sqrt{3}, - \frac{1}{2})$

$Q \equiv (- a e, - \frac{b^{2}}{a}) = (- \sqrt{3}, - \frac{1}{2})$ $∴$ $P Q = 2 \sqrt{3}$

$Now if P Q is the length of latus rectum of the parabola whose equation is to be found, then$

$P Q = 4 a \Rightarrow 2 \sqrt{3} = 4 a \Rightarrow a = \frac{\sqrt{3}}{2}$

Also as $P Q$ is horizontal, parabola with $P Q$ as latus rectum can be upward parabola (with vertex at A) or downward parabola (with vertex at $A'$ ) as shown in the figure.

For upward parabola,

$A R = a = \frac{\sqrt{3}}{2},$ $∴$ $Coordinates of A = (0, - (\frac{\sqrt{3} + 1}{2}))$

$∴$ $Equation of upward parabola is$

$x^{2} = 2 \sqrt{3} (y + \frac{\sqrt{3} + 1}{2})$ $\Rightarrow x^{2} - 2 \sqrt{3} y = 3 + \sqrt{3} \dots (i)$

$For downward parabola$ $A' R = a = \frac{\sqrt{3}}{2}$

$∴$ $Coordinates of A' = (0, - (\frac{1 - \sqrt{3}}{2}))$

$∴$ $Equation of downward parabola is given by$

$x^{2} = - 2 \sqrt{3} (y + \frac{1 - \sqrt{3}}{2})$ $\Rightarrow x^{2} + 2 \sqrt{3} y = 3 - \sqrt{3} \dots (i i)$

$∴$ $Equation of required parabola is given by equation (i) or (i i)$

Solution

Q.
Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ , where $y_{1} < 0, y_{2} < 0$ , be the end points of the latus rectum of the ellipse $x^{2} + 4 y^{2} = 4 .$ The equations of parabolas with latus rectum $P Q$ are [2008]

1 $x^{2} + 2 \sqrt{3} y = 3 + \sqrt{3}$

2 $x^{2} - 2 \sqrt{3} y = 3 + \sqrt{3}$

3 $x^{2} + 2 \sqrt{3} y = 3 - \sqrt{3}$

4 $x^{2} - 2 \sqrt{3} y = 3 - \sqrt{3}$

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Solution

Q. Let P(x1,y1) and Q(x2,y2), where y1<0, y2<0, be the end points of the latus rectum of the ellipse x2+4y2=4. The equations of parabolas with latus rectum PQ are [2008]

1 x2+23y=3+3

2 x2-23y=3+3

3 x2+23y=3-3