First slide

Equation of line in Straight lines

Question

The perpendicular distance from the point $(1, 1)$ to the line which is perpendicular to $4 x - 3 y + 1 = 0$ and passing through $(3, 5)$ is

Moderate

A

$\frac{22}{3}$
B

$\frac{21}{5}$
C

$\frac{22}{\sqrt{5}}$
D

$\frac{22}{5}$

Solution

$The equation of any line perpendicular to the line 4 x - 3 y + 1 = 0 can be taken as$ $3 x + 4 y + k = 0$
$It is passing through the point (3, 5)$
It implies
$\begin{matrix} 3 (3) + 4 (5) + k = 0 \\ 9 + 20 + k = 0 \\ k = - 29 \end{matrix}$
$Hence the equation of the line perpendicular to the line 4 x - 3 y + 1 = 0 and passing through$
$the point (3, 5) is 3 x + 4 y - 29 = 0$
$The perpendicular distance from a point P (x_{1}, y_{1}) to the line a x + b y + c = 0 is |\frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}}|$
$The perpendicular distance from a point (1, 1) to the line 3 x + 4 y - 29 = 0 is$ $|\frac{3 (1) + 4 (1) - 29}{\sqrt{16 + 9}}| = \frac{22}{5}$

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