Question

A rod $A B$ of length $15\, cm$ rests in between two coordinate axes in such a way that the end point A lies on $x$ -axis and end point B lies on y-axis. A point $P ( x , y )$ is taken on the rod in such a way that $AP =6\, cm$. Then, the locus of $P$ is $a / an$.

• A. circle
• B. ellipse
• C. parabola
• D. hyperbola

Answer 1

Answer: (B)

Let $AB$ be the rod making an angle $\theta$ with $OX$ as shown in figure and $P ( x , y )$ the point on it such that $AP =6\, cm$.
Since, $AB =15\, cm$, we have $PB =9 \,cm$

From $P$, draw $PQ$ and $PR$ perpendiculars on $y$-axis and $x$-axis, respectively.
From $\Delta PBQ , \cos \theta=\frac{ x }{9}$
From $\Delta PRA , \sin \theta=\frac{ y }{6}$
Since, $\cos ^{2} \theta+\sin ^{2} \theta=1$
$\left(\frac{x}{9}\right)^{2}+\left(\frac{y}{6}\right)^{2}=1$ or $\frac{x^{2}}{81}+\frac{y^{2}}{36}=1$
Thus, the locus of $P$ is an ellipse.