Question

A line of fixed length $a+b$ moves so that its ends are always on two fixed perpendicular straight lines. Then the locus of the point which divides this line into portions of lengths a and b is $a/an$

• A. ellipse
• B. parabola
• C. straight line
• D. none of these

Answer 1

Answer: (A)

Let $AB$ be the line.
Let $AP=a$ and $PB=b$, so the $AB=a+b$.
If $AB$ makes an angle $\theta$ with the $x$-axis and the coordinates of $P$ are $(x,y)$, then
in $\Delta APL,x=a\cos \theta$
in $\Delta PBQ,y=b\sin \theta$
Therefore, the locus of $P(x,y)$ is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
which is an ellipse.