Question

The equation of a line inclined at an angle $\frac{\pi }{4}$ to the $x$-axis, such that the two circles $x^2+y^2=4,x^2+y^2-10x-14y+65=0$ intercept equal lengths on it is

• A. $2x-2y-3=0$
• B. $2x-2y+3=0$
• C. $x-y+6=0$
• D. $x-y-6=0$

Answer 1

Answer: (A)

$2x-2y-3=0$

Let the equation of the required line be $y=x+c$ or $y-x=c$

Perpendicular from $(0,0)$ on line (i) is $\left|\frac{-c}{\sqrt{2}}\right|=\frac{c}{\sqrt{2}}$

In $\Delta AON,\sqrt{2^2-{\left(\frac{c}{\sqrt{2}}\right)}^2}=AN$

and in $\Delta CPM,\sqrt{3^2-{\left(\frac{2-c}{\sqrt{2}}\right)}^2}=CM$

Given $AN=CM\Rightarrow 4-\frac{c^2}{2}=9-\frac{(2-c{)}^2}{2}$

$\Rightarrow c=-\frac{3}{2}$

Therefore, the equation of line $y=x-\frac{3}{2}$ or $2x-2y-3=0$.