The equation of the line inclined at an angle of `pi/4` to the x-axis ,such that the two circles `x^2+y^2=4` and `x^2+y^2 -10 x-14 y+65=0` intercept equal length on it, is (A) `2x-2y-3=0` (B) `2x-2y+3=0` (C) `x-y+6=0` (D) `x-y-6=0`

Options

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

Answer

1

Explanation

Let the equation of line be
`y=x+c` or `y-x=c`
Ther perpendicular from (0,0) on line `(1)` in `|c//sqrt(2)|`
The perpendicular from `(5,7)` on line (1) is `|c-2|//sqrt(2)`
In `Delta AON`,
`sqrt(2^(2)-((c )/(sqrt(2)))^(2))=AN`
and in `Delta CPM`
`sqrt(3^(2)-((2-c)/(sqrt(2)))^(2))=CM`
Given `AN=CM`
or `4-(c^(2))/(2)=9=9-((2-c)^(2))/(2)` or `c= -(3)/(2)`
Therefore, the equation of the line is
`y=x-(3)/(2)` or `2x-2y-3=0`