The normal at a point on the ellipse meets the -axis at . If is the midpoint of the line segment , then the locus of intersects the latus rectums of the given ellipse at the points [2009]

Question

The normal at a point $P$ on the ellipse $x^{2} + 4 y^{2} = 16$ meets the $x$ -axis at $Q$ . If $M$ is the midpoint of the line segment $P Q$ , then the locus of $M$ intersects the latus rectums of the given ellipse at the points [2009]

Smart Achievers · Accepted Answer

(3)

$Given ellipse is \frac{x^{2}}{4^{2}} + \frac{y^{2}}{2^{2}} = 1$

$P Q$ $M$

$Let P (4 \cos θ, 2 \sin θ) be any point on the ellipse, then equation of normal at P is$

$4 x \sin θ - 2 y \cos θ = 12 \sin θ \cos θ$

$\Rightarrow \frac{x}{3 \cos θ} - \frac{y}{6 \sin θ} = 1$

$∴$ $Q, the point where normal at P meets x-axis, has coordinates (3 \cos θ, 0)$

$∴$ $Mid point of P Q is M (\frac{7 \cos θ}{2}, \sin θ)$

$For locus of point M we consider$

$x = \frac{7 \cos θ}{2}, y = \sin θ \Rightarrow \cos θ = \frac{2 x}{7}, \sin θ = y$

$Since \sin^{2} θ + \cos^{2} θ = 1$

$∴$ $\frac{4 x^{2}}{49} + y^{2} = 1 . . . (i)$

$Also the latus rectum of given ellipse is$

$x = \pm a e = \pm 4 \times \frac{\sqrt{3}}{2} \Rightarrow x = \pm 2 \sqrt{3} . . . (i i)$

$On solving equations (i) and (ii), we get$

$\frac{4 \times 12}{49} + y^{2} = 1 \Rightarrow y^{2} = \frac{1}{49} or y = \pm \frac{1}{7}$

$∴$ $The required points are (\pm 2 \sqrt{3}, \pm \frac{1}{7}) .$

Solution

Q.
The normal at a point $P$ on the ellipse $x^{2} + 4 y^{2} = 16$ meets the $x$ -axis at $Q$ . If $M$ is the midpoint of the line segment $P Q$ , then the locus of $M$ intersects the latus rectums of the given ellipse at the points [2009]

1 $(\pm \frac{3 \sqrt{5}}{2}, \pm \frac{2}{7})$

2 $(\pm \frac{3 \sqrt{5}}{2}, \pm \sqrt{\frac{19}{4}})$

3 $(\pm 2 \sqrt{3}, \pm \frac{1}{7})$

4 $(\pm 2 \sqrt{3}, \pm \frac{4 \sqrt{3}}{7})$

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Solution

Q. The normal at a point P on the ellipse x2+4y2=16 meets the x-axis at Q. If M is the midpoint of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points [2009]

1 (±352,±27)

2 (±352,±194)

3 (±23,±17)