STATEMENT- 1 : The circles $x^{2} + y^{2} - 2 x - 6 y + 3 = 0 and x^{2} + y^{2} - 6 x - 2 y + 9 = 0$ cut each other orthogonally.

and

STATEMENT-2 : If two intersecting circles $x^{2} + y^{2} + 2 g_{1} + 2 f_{1} y + c_{1} = 0 and x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0$ are orthogonal then $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

statement-1 is True, statement-2 is True, statement-2 is a correct explanation for statement-1

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statement-1 is True, statement-2 is True; statement-2 is NOT a correct explanation of statement-1

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statement-1 is True, statement-2 is True; statement-2 is a correct explanation for statement-1

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statement-1 is False, statement-2 is True

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Solution

The correct option is C statement-1 is True, statement-2 is True; statement-2 is a correct explanation for statement-1
$g - 2 = - 1, f_{1} = - 3, c - 1 = 3; g_{2} = - 3, f_{2} = - 1, c_{2} = 9$

$2 g_{1} g_{2} = 2 f_{1} f - 2 = 6 + 6 = 12$

$c_{1} + c_{2} = 9 + 3 = 12 = 2 g - 1 g_{2} + 2 f_{1} f_{2}$ .

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Similar questions

Two circles $x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0 a n d x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c^{2} = 0$ are said to be orthogonal. Then $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$

Q. Prove that the circle through the origin and cutting the circles

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0

and

x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0

orthogonally is

∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + y^{2} & - x & - y c_{1} & g_{1} & f_{1} c_{2} & g_{2} & f_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

Q. Find the equation of a circle passing through the origin and cutting the circles

x^{2} + y^{2} + 2 g_{1} x + 2 f y_{1} + c_{1} = 0

and

x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0

orthogonally.

Q. Assertion :The equation of the circle passing through the point

(2 a, 0)

and whose radical axis is

x = \frac{a}{2}

with respect to the circle

x^{2} + y^{2} = a^{2}

, will be

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

Reason: The equation of radical axis of two circles

x^{2} + y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0

and

x^{2} + y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0

2 (g_{1} - g_{2}) x + 2 (f_{1} - f_{2}) y + (c_{1} - c_{2}) = 0

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STATEMENT- 1 : The circles x2+y2−2x−6y+3=0 and x2+y2−6x−2y+9=0 cut each other orthogonally. and STATEMENT-2 : If two intersecting circles x2+y2+2g1+2f1y+c1=0 and x2+y2+2g2x+2f2y+c2=0 are orthogonal then 2g1g2+2f1f2=c1+c2

The correct option is C statement-1 is True, statement-2 is True; statement-2 is a correct explanation for statement-1g−2=−1, f1=−3, c−1=3; g2=−3, f2=−1, c2=9 2g1g2=2f1f−2=6+6=12 c1+c2=9+3=12=2g−1g2+2f1f2.