TS EAMCET · Maths · Circle

The equation of the circle passing through $(1, 1)$ and through the points of intersection of the circles $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is

A $4 x^{2} + 4 y^{2} + 30 x - 13 y - 25 = 0$
B $2 x^{2} + 2 y^{2} + 15 x - 19 y = 0$
C $4 x^{2} + 4 y^{2} + 25 x + 12 y - 45 = 0$
D $4 x^{2} + 4 y^{2} + 13 x - 30 y + 9 = 0$

Verified Solution

Answer & Solution

Correct Answer

(A) $4 x^{2} + 4 y^{2} + 30 x - 13 y - 25 = 0$

Step-by-step Solution

Detailed explanation

Using family of circles, let the equation of the circle passing through the points of intersection of the circles x2+y2+13x-3y=0 and 2x2+2y2+4x-7y-25=0 be 2x2+2y2+4x-7y-25+λx2+y2+13x-3y=0 Since it passes through 1,1, so 212+212+41-71-25+λ12+12+131-31=0 ⇒λ=2…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The equation of the circle passing through 1, 1 and through the points of intersection of the circles x2+y2+13x-3y=0 and 2x2+2y2+4x-7y-25=0 is

Step-by-step Solution

The equation of the circle passing through $(1, 1)$ and through the points of intersection of the circles $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is