Let a circle C touch the lines L1:4x−3y+K1=0 and L2:4x−3y+K2=0, K1,K2∈R. If a line passing through the centre of C intersects L1 at (−1,2) and L2 at (3,−6), then the equation of the circle C is:
Solutions · 1
Official solution
Co-ordinate of centre \[C \equiv \left( {{{3 + ( - 1)} \over 2},{{ - 6 - 2} \over 2}} \right) \equiv (1, - 2)\]
L1 is passing through A
\[ \Rightarrow - 4 - 6 + {K_1} = 0\]
\[ \Rightarrow {K_1} = 10\]
L2 is passing through B
\[ \Rightarrow 12 + 18 + {K_2} = 0\]
\[ \Rightarrow {K_2} = - 30\]
Equation of L1:4x−3y+10=0
Equation of L2=4x−3y−30=0
Diameter of circle \[ = \left| {{{10 + 30} \over {\sqrt {{4^2} + {{( - 3)}^2}} }}} \right| = 8\]
\[\Rightarrow\] Radius = 4
Equation of circle (x−1)2+(y+2)2=16