Q.441. Tangents - Co-ordinate geometry.

Question 441.
If  from  any point  on  the  common  chord  of  two  intersecting  circles,  tangents  are  drawn  to  the circles,  prove  that they are of  equal  lengths.

Answer 441.
Let x^2 + y^2 + 2g1x + 2f1y + c1 = 0
and x^2 + y^2 + 2g2x + 2f2y + c2 = 0 be any two circles.
Their common chord is obtained by taking the difference of the equations and is
2(g1 - g2)x + 2(f1 - f2)y + c1 - c2 = 0
Let (h, k) be any point external to the two circles on the above common chord.
=> 2(g1 - g2)h + 2(f1 - f2)k + c1 - c2 = 0 ... ( 1 )

If L1 and L2 are the lengths of tangents from (h, k) to the given circles,
L1^2 = h^2 + k^2 + 2g1h + 2f1k + c1 and
L2^2 = h^2 + k^2 + 2g2h + 2f2k + c2
=> L1^2 - L2^2
= 2(g1 - g2)h + 2(f1 - f2)k + c1 - c2
= 0 ... [from eqn. ( 1 )]
=> L1^2 = L2^2
=> L1 = L2.

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