The locus of the point of intersection of the tangents at the extremities of a chord of the circle x2 + y2 = a2, which touches the circle x2 + y2 – 2ax = 0, is
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y2 = a(a – 2x)
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x2 = a(a – 2y)
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x2 + y2 = (x + a)2
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x2 + y2 = (y + a)2
The chord of contact of tangents from (h, k) to x^2 + y^2 = a^2 is hx + ky = a^2. This chord touches x^2 + y^2 - 2ax = 0 (center (a, 0), radius a). The distance from (a, 0) to the line hx + ky - a^2 = 0 is |ha - a^2| / sqrt(h^2 + k^2) = a. Squaring gives (a(h-a))^2 = a^2(h^2 + k^2), which simplifies to (h-a)^2 = h^2 + k^2, or h^2 - 2ah + a^2 = h^2 + k^2. Thus k^2 = a^2 - 2ah. Replacing (h, k) with (x, y) gives y^2 = a(a - 2x).
The locus of the point of intersection of tangents at the extremities of a chord is the polar of the center of the given circle. Let the point of intersection be P(x1, y1). The chord of contact of tangents drawn from P to the circle x squared plus y squared equals a squared is x x1 plus y y1 equals a squared. This line touches the circle x squared plus y squared minus 2ax equals 0, whose center is (a, 0) and radius is a. Using the condition that the distance from the center to the tangent line equals the radius, we equate the distance from (a, 0) to x x1 + y y1 - a squared = 0 with a. This yields (a x1 - a squared) divided by the square root of (x1 squared + y1 squared) equals a. Squaring both sides gives 2 a squared x1 equals a squared y1 squared. Replacing x1 and y1 with x and y gives the locus y squared equals a(a - 2x).