Question

If the extremities of a line segment of length $l$ move in two fixed perpendicular straight lines, then the locus of that point which divides this lines segment in the ratio $1:2$ is

• A. a parabola
• B. an ellipse
• C. a hyperbola
• D. None of these

Answer 1

Answer: (B)

an ellipse

Let the two fixed $\perp$ straight lines be the coordinates axes.

Let $P(h,k)$ be the point whose locus is required.

Let $PA:PB=1:2$

Then $PA=\frac{l}{3}$ and $PB=\frac{2l}{3}$

$k=\frac{l}{3}\sin \theta \Rightarrow 3k=l\sin \theta$ ...(1)

and $h=\frac{2l}{3}\cos \theta \Rightarrow \frac{3h}{2}=l\cos \theta$ ...(2)

Squaring and adding (1) and (2), we get

$9k^2+\frac{9h^2}{4}=l^2\Rightarrow 9h^2+36k^2=4l^2$

$\therefore$ Locus of $P(h,k)$ is $9x^2+36y^2=4l^2$, which is an ellipse.