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Q.

Given equation represents a circle $a x^{2} + b y^{2} + 2 h x y + 2 x - 4 y - 3 = 0$ and use notations as given below $S : a x^{2} + b y^{2} + 2 h x y + 2 x - 4 y - 3$ , w.r.t point $P (x_{1}, y_{1}),$ $S_{1} : a x x_{1} + b y y_{1} + h (x_{1} y + x y_{1}) + x + x_{1} - 2 (y + y_{1}) - 3$ ,w.r.t point $Q (x_{2}, y_{2})$ , $S_{22} : a x_{2}^{2} + b y_{2}^{2} + 2 h x_{2} y_{2} + 2 x_{2} - 4 y_{2} - 3$ then, which of the following option(s) must be correct?

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a

$h = 0$

b

Equation of tangent at point $P (x_{1}, y_{1}) o n S = 0 i s S_{1} = 0$

c

The condition for the point $Q (x_{2}, y_{2})$ to lie inside of the given circle $S = 0 i s S_{22} < 0$

d

$a = b$

answer is A, B, C.

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Basic formulas of circles

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Given equation represents a circle ax2+by2+2hxy+2x−4y−3=0 and use notations as given below S:ax2+by2+2hxy+2x−4y−3, w.r.t point P(x1,y1), S1:axx1+byy1+h(x1y+xy1)+x+x1−2(y+y1)−3,w.r.t point Q(x2,y2), S22:ax22+by22+2hx2y2+2x2−4y2−3 then, which of the following option(s) must be correct?