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Question
Show that x = −1 is a tangent to circle x2 + y2 − 2y = 0 at (−1, 1).
Solution
We have to find the equation of tangent to the circle x2 + y2 – 2y = 0 at (– 1, 1).
The equation of the tangent to the circle
x2 + y2 + 2gx + 2fy + c = 0 at (x1, y1) is
xx1 +yy1 + g(x + x1) + f(y + y1) + c = 0
Comparing the equation x2 + y2 – 2y = 0 with
x2 + y2 + 2gx + 2fy + c = 0, we get,
g = 0, f = – 1, c = 0
∴ The equation of the tangent to the given circle at (– 1, 1) is
x(–1) + y(1) – 0(x – 1) – 1(y + 1) + 0 = 0
∴ –x + y – y – 1 = 0
∴ x = – 1
Hence, x = – 1 is tangent to the circle x2 + y2 – 2y = 0 at (– 1, 1).
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