Complex Numbers Question - None

Question

Let P and Q be two points on the circle $|w| = r$ represented by $w_1$ and $w_2$ respectively. Then the complex number representing the point of intersection of the tangents at P and Q is:

Accepted Answer

Step 1: Set up the tangent condition

For circle |w| = r, the tangent at point w₁ is perpendicular to the radius Ow₁. The equation of tangent at w₁ is: Re(w·w̄₁) = r²

Step 2: Write tangent equations at both points

Tangent at P(w₁): Re(w·w̄₁) = r²

Tangent at Q(w₂): Re(w·w̄₂) = r²

Step 3: Express in expanded form

These can be written as: w·w̄₁ + w̄·w₁ = 2r²

And: w·w̄₂ + w̄·w₂ = 2r²

Step 4: Use the pole-polar relationship

For a circle |w| = r, if S is the intersection of tangents at w₁ and w₂, then S is the pole of the chord w₁w₂. The pole-polar relationship gives: S = r²/(w̄₁w̄₂/w₁w₂)·(w₁w₂)/(w₁+w₂)

Step 5: Apply the direct formula

For circle |w| = r with tangents at w₁ and w₂, the point of intersection is given by the formula derived from the harmonic conjugate property:

S = 2w₁w₂/(w₁ + w₂)

Step 6: Verify dimensions

Since w₁ and w₂ lie on |w| = r, we have |w₁| = |w₂| = r. The expression 2w₁w₂/(w₁+w₂) is dimensionally consistent and represents a point whose distance from origin is r²/|w₁+w₂| times 2, which is characteristic of the intersection of tangents.

∴ Answer: C

Question:

Step-by-Step Solution

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