Find the equation of pair of tangenst drawn to circle `x^(2)+y^(2)-2x+4y-4=0` from point P(-2,3). Also find the angle between tangest.

Explanation

We have circle `x^(2)+y^(2)-2x+4y-4=0` or `S=0`.
Tangents are drawn from point P(-2,3) to the circle.
Equation of pair of tangents is
`T^(2)=S S_(1)`
or `(-2x+3y-(x-2)+2(y+3)-4)^(2)`
`=(x^(2)+y^(2)-2x+4y-4)((-2)^(2)+3^(2)-2(-2)+4(3)-4`
`implies (-3x+5y+4)^(2)=25(x^(2)+y^(2)-2x+4y-4)`
`implies 16x^(2)+30xy-26x+60y-116=0`
`implies 8x^(2)+15xy-13x+30y-58=0`
Here, a=8, b=0 and `h=15//2`
If angle between tangents is `theta`, then
`tan theta=(2 sqrt(h^(2)-ab))/(|a+b|)=(2sqrt((225)/(4)-0))/(|8+0|)=(15)/(8)`
`:. theta =tan ^(-1).(15)/(8)`