First slide

Equation of line in Straight lines

Question

A line is drawn from the point $P (α, β)$ making an angle $θ$ with the positive direction of x-axis, to meet the line ax + by + c = 0 at Q. The length of PQ is

Moderate

A

$- \frac{a α + b β + c}{a \cos θ + b \sin θ}$
B

$|\frac{a α + b β + c}{\sqrt{a^{2} + b^{2}}}|$
C

$\frac{a α + b β + c}{a \cos θ + b \sin θ}$
D

none of these

Solution

Equation of a straight line passing through the point $P (α, β)$ and making an angle $θ$ with positive direction of x-axis is
$\frac{x - α}{\cos θ} = \frac{y - β}{\sin θ} = r (s a y)$
Coordinates of any point on this line are
$(α + r \cos θ, β + r \sin θ)$
If it lies on the line ax + by + c = 0, then
$a (α + r \cos θ) + b (β + r \sin θ) + c = 0$
$\Rightarrow r = - \frac{a α + b β + c}{a \cos θ + b \sin θ}$
Thus, $P Q = r = - \frac{a α + b β + c}{a \cos θ + b \sin θ}$

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