Question

A rod $AB$ of length $15cm$ 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 $AP=6cm$. 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 $AP=6cm$.
Since, $AB=15cm$, we have $PB=9cm$

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.