Question

Number of points having distance $\sqrt{5}$ from the straight line $x-2y+1=0$ and a distance $\sqrt{13}$ from the line $2x+3y-1=0$ is

• A. $1$
• B. $2$
• C. $4$
• D. $5$

Answer 1

The correct option is C $4$

Let us consider the required points be $(h,k)$.
Their distance from the straight line $x-2y+1=0$ is given as $\sqrt{5}$
$\Rightarrow \frac{|h-2k+1|}{\sqrt{5}}=\sqrt{5}\Rightarrow h-2k+1=5\cdots \left(1\right)$
and $h-2k+1=-5\cdots \left(2\right)$

Their distance from the straight line $2x+3y-1=0$ is given as $\sqrt{13}$
$\Rightarrow \frac{|2h+3k-1|}{\sqrt{13}}=\sqrt{13}\Rightarrow 2h+3k-1=13\cdots \left(3\right)$
and $2h+3k-1=-13\cdots \left(4\right)$

On solving $\left(1\right)$ with $\left(3\right)$ and $\left(4\right)$, we get two points.

On solving $\left(2\right)$ with $\left(3\right)$ and $\left(4\right)$, we get two points.
So, we will get four different values of $(h,k)$
Therefore, four points satisfy the required relation.