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 $\left(- 9,1\right)$