First slide

Equation of line in Straight lines

Question

The distance of the line $x + y - 8 = 0$ from $(4, 1)$ measured along the direction whose slope is $- 2$ , is

Moderate

A

$3 \sqrt{5}$
B

$6 \sqrt{5}$
C

$2 \sqrt{5}$
D

none of these

Solution

The equation of a line passing through $P (4, 1)$ and slope $- 2$ is
$\frac{x - 4}{\frac{1}{\sqrt{5}}} = \frac{y - 1}{\frac{2}{\sqrt{5}}} [∵ \tan θ = - 2 \Rightarrow \cos θ = - \frac{1}{\sqrt{5}}, \sin θ = \frac{2}{\sqrt{5}}]$
Suppose it cuts $x + y - B = 0$ at $Q$ such that $P Q = r$ . Then, the coordinates of $Q$ are given by
$\frac{x - 4}{- \frac{1}{\sqrt{5}}} = \frac{y - 1}{2} = r \Rightarrow x = 4 - \frac{r}{\sqrt{5}}, y = 1 + \frac{2 r}{\sqrt{5}}$
Since $Q$ lies on the line $x + y - B = 0 .$
$∴ 4 - \frac{r}{\sqrt{5}} + 1 + \frac{2 r}{\sqrt{5}} - 8 = 0 \Rightarrow r = 3 \sqrt{5}$
Hence, required distance $= 3 \sqrt{5}$ units.

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