If the tangents are drawn to the circle `x^2+y^2=12` at the point where it meets the circle `x^2+y^2-5x+3y-2=0,` then find the point of intersection of these tangents.

Explanation

We have circles
`S_(1)-= x^(2)+y^(2)-12=0` (1)
and `S_(2)-=x^(2)+y^(2)-5x+3y-2=0` (2)

Circles are intersecting at point A and B.
Tangents are drawn to circle `S_(1)=0` at A and B, which meet at point P(h,k).
AB is common chord of the circles, whose equation is `S_(1)-S_(2)=0`.
or `5X-3Y-10=0`. (3)
Also, line AB is chord of contact of the circle `S_(1)=0`, w.r.t. point P.
`:. ` Equation of AB is
`hx+ky-12=0` (4)
Equations (3) and (4) represents the same straight line.
`:. (h)/(5)=(k)/(-3)=(-12)/(-10)`
`implies h=6,k=-18//5`
Hence, the required point is `P(6,-18//5)` .
In the figure, `C_(1)M=(|5(0)-3(0)-10|)/(sqrt(5^(2)+(-3)^(2)))=(10)/(sqrt(34))`
In triangle `C_(1)MA,AM^(2)=C_(1)A^(2)-C_(1)M^(2)-12-(100)/(34)=(154)/(17)`
`:. `Length of common chord , `AB=2AM=2sqrt((154)/(17))`