Question

A rod $AB$ of the length $30$ units slips on the coordinate axes such that $A$ lies on $x-$axis and $B$ lies on $y-$axis. Then, the locus of $C$ if $BC=2AC$ is

• A. $x^2+4y^2=400$
• B. $4x^2+y^2=400$
• C. $x^2+4y^2=100$
• D. $4x^2+y^2=100$

Answer 1

The correct option is A $x^2+4y^2=400$

Let the points are $A(a,0)$ and $B(0,b)$
$\therefore \sqrt{a^2+b^2}=30\cdots \left(1\right)$
Let the coordinates of $C$ is $C(h,k)$
$\because C$ divides the rod $AB$ into $2:1$ ratio
$\therefore h=\frac{2a}{3}\Rightarrow a=\frac{3h}{2}k=\frac{b}{3}\Rightarrow b=3k$
From $\left(1\right)$
$9k^2+\frac{9h^2}{4}={30}^2{h}^2+4k^2=400$
locus of $C$ is
$x^2+4y^2=400$