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Equation of Normal at a Point (x,y) in Terms of f'(x)

A 1,0 and B 0...

Question

A(1, 0) and B(0, 1) are two fixed points on the circle $x^{2} + y^{2} = 1$ . C is a variable point of this circle. As C moves, the locus of orthocenter of the triangle ABC is

$x^{2} + y^{2} - 2 x - 2 y + 1 = 0$

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$x^{2} + y^{2} - x - y = 0$

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$x^{2} + y^{2} = 4$

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$x^{2} + y^{2} + 2 x - 2 y + 1 = 0$

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Solution

The correct option is A
$x^{2} + y^{2} - 2 x - 2 y + 1 = 0$

Centroid divide the line joining orthocenter and circumcentre in 2 : 1

$\Rightarrow h = 1 + c o s θ, k = 1 + s i n θ \Rightarrow (x - 1)^{2} + (y - 1)^{2} = 1$

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From origin, chords are drawn to the circle $x^{2} + y^{2} - 2 y = 0$ .The locus of the middle points of these chords is

x^{2} + y^{2} - 2 x - 4 y + 4 = 0

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