Question

If $P(x,y)$ divides the line segment joining $A(5,0)$ and $B(10\cos \theta ,10\sin \theta )$ internally in the ratio $2:3,$ then which of the following is (are) true?

• A. Distance of $P$ from $(3,0)$ is always constant.
• B. $x\in [0,3]$
• C. $y\in [-4,0]$
• D. $x^2+y^2-6x-7=0$

Answer 1

Answer: (A), (D)

The correct options are
A Distance of $P$ from $(3,0)$ is always constant.
D $x^2+y^2-6x-7=0$

By section formula,
$(x,y)=(\frac{20\cos \theta +15}{5},\frac{20\sin \theta +0}{5})$
$=(4\cos \theta +3,4\sin \theta )$
$\Rightarrow x=4\cos \theta +3,y=4\sin \theta$

Since, $\cos \theta$ and $\sin \theta \in [-1,1]$
$\Rightarrow x\in [-1,7]$ and $y\in [-4,4]$

Distance of $P$ from $(3,0)$ is $\sqrt{(x-3{)}^2+y^2}$ $=\sqrt{16{\sin }^2\theta +16{\cos }^2\theta }=4,$ which is a constant.
$\Rightarrow (x-3{)}^2+y^2=4^2$
$\Rightarrow x^2+y^2-6x-7=0$