Multiple choice

If points represented by the complex numbers a,b,c lie on a circle with centre O and radius r. The tangent at c cuts the chord joining the points a,b at z. Then the value of $\frac { { a }^{ -1 }+{ b }^{ -1 }-2{ c }^{ -1 } }{ { a }^{ -1 }{ b }^{ -1 }-{ c }^{ -2 } } $.

  1. z

  2. 2z

  3. 3z

  4. none of these
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A Correct answer
Explanation

This is a known property in complex geometry where the tangent at a point on a circle relates to the inversion of points. The expression simplifies to the complex number z representing the intersection point.

AI explanation

By geometric properties of the circle, if z lies on the tangent at c and on the chord ab, it satisfies the power of a point theorem. The given algebraic expression simplifies to z because the points a, b, and c lie on the circle centered at O. Rearranging the terms verifies the identity equals z.

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