First slide

Equation of line in Straight lines

Question

line through the point $A (2, 4)$ intersects the line $x + y = 9$ at the point $P .$ The minimum distance of $A P,$ is

Moderate

A

$\frac{5}{\sqrt{2}}$
B

$\frac{7}{\sqrt{2}}$
C

$\frac{3}{\sqrt{2}}$
D

$\frac{1}{\sqrt{2}}$

Solution

The equation of a line passing through $A (2, 4)$ is $\frac{x - 2}{\cos θ} = \frac{y - 4}{\sin θ}$
Suppose it cuts the line $x + y = 9$ at point $P$ whose coordinates are given by
$\frac{x - 2}{\cos θ} = \frac{y - 4}{\sin θ} = r$ i.e. $x = 2 + r \cos θ, y = 4 + r \sin θ$
$\begin{array}{ll} ∴ 2 + r \cos θ + 4 + r \sin θ = 9 \\ \Rightarrow r (\cos θ + \sin θ) = 3 \\ \Rightarrow r = \frac{3}{\cos θ + \sin θ} \end{array}$
$\Rightarrow r \geq \frac{3}{\sqrt{2}} [∵ \cos θ + \sin θ \leq \sqrt{2} ∴ \frac{1}{\cos θ + \sin θ} \geq \frac{1}{\sqrt{2}}]$
Hence, the minimum value of $A P$ is $\frac{3}{\sqrt{2}}$

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