The chord of contact of tangents from a point P to a circle passes through Q. If L1 and L2 are the lengths of tangent from P and Q to the circle, then PQ is equal to..??

Dear Chirag,

Suppose we have the general equation of a conic curve given by

The equation of the chord joining any two points,  and , on the curve may be written

.		.
.		.

as the equation is clearly first-degree in x and y (the terms above first-degree cancel out), and it is satisfied by the two points  and . Putting  and , we get the equation of the tangent line

or, expanding,

Adding  to both sides will cause the right-hand side to vanish, because  satisfies the equation of the curve. Thus the equation of the tangent becomes

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