Question

A line passes through $(2,3)$ and the portion of it intercepted between co-ordinate axes is divided in the ratio $2:3$ at the point $(2,3).$ Then equation of the line in Normal form is

• A. $x\cos \frac{\pi }{4}+y\sin \frac{\pi }{4}=\frac{5}{\sqrt{2}}$
• B. $x\cos \frac{\pi }{3}+y\sin \frac{\pi }{3}=\frac{3}{\sqrt{2}}$
• C. $\frac{9}{\sqrt{97}}x+\frac{4}{\sqrt{97}}y=\frac{30}{\sqrt{97}}$
• D. $\frac{3}{\sqrt{13}}x+\frac{2}{\sqrt{13}}y=\frac{6}{\sqrt{13}}$

Answer 1

Answer: (A), (C)

The correct options are
A $x\cos \frac{\pi }{4}+y\sin \frac{\pi }{4}=\frac{5}{\sqrt{2}}$
C $\frac{9}{\sqrt{97}}x+\frac{4}{\sqrt{97}}y=\frac{30}{\sqrt{97}}$


Let $A(h,0)$ and $B(0,k)$ are the end points of the line sagment.

Case $1:AB$ is divided in the ratio $2:3$ at $(2,3)$
$\therefore (\frac{3h}{5},\frac{2k}{5})=(2,3)$
$\Rightarrow h=\frac{10}{3},k=\frac{15}{2}$

Equation of line is $\frac{3x}{10}+\frac{2y}{15}=1$
$\Rightarrow 9x+4y=30$
$\therefore$ Normal form is $\frac{9}{\sqrt{97}}x+\frac{4}{\sqrt{97}}y=\frac{30}{\sqrt{97}}$

Case $2:AB$ is divided in the ratio $3:2$ at $(2,3)$
$\therefore (\frac{2h}{5},\frac{3k}{5})=(2,3)$
$\Rightarrow h=5,k=5$

Equation of line is $x+y=5$

$\therefore$ Normal form is $\frac{1}{\sqrt{2}}x+\frac{1}{\sqrt{2}}y=\frac{5}{\sqrt{2}}$
$\Rightarrow x\cos \frac{\pi }{4}+y\sin \frac{\pi }{4}=\frac{5}{\sqrt{2}}$