Question

Point $P\left(-1,7\right)$ lies on the line $4x+3y=17.$ Then the coordinates of the points farthest from the line which are at a distance of $10$ units from the point $P$ are

• A. $\left(7,13\right)$ and $\left(-9,1\right)$
• B. $\left(5,15\right)$ and $\left(-1,-7\right)$
• C. $\left(-1,5\right)$ and $\left(15,-7\right)$
• D. $\left(15,5\right)$ and $\left(-7,-1\right)$

Answer 1

Answer: (A)

There are two points $A$ & $B$ which are at a distance of $10$ units from $P$ and farthest ( $10$ unit distance) from the line $4x+3y=17.$
The slope of $AB=\frac{3}{4}$ {Because the slope of $4x+3y=17$ is $-\frac{4}{3}$ }
Let line $AB$ makes an angle $\theta$ with $x$ -axis then $tan\theta =\frac{3}{4}\Rightarrow \left(\cos \theta ,\sin \theta \right)=\left(\frac{4}{5},\frac{3}{5}\right)$
$\Rightarrow A$ or $B=\left(10\cos \theta -1,10\sin \theta +7\right)$
or $\Rightarrow A$ or $B=\left(-10\cos \theta -1,-10\sin \theta +7\right)$
$\Rightarrow A$ or $B=\left(7,13\right)$ or $\left(-9,1\right)$