AP EAMCET · Maths · Circle

If $P (x_{1}, y_{1})$ is a point such that the lengths of the tangents from it to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ are in the ratio $2 : 3$ , then the locus of $P$ is

A $x^{2} + y^{2} + 24 x - 36 y + 62 = 0$
B $x^{2} + y^{2} - 12 x - \frac{126}{5} y - \frac{212}{5} = 0$
C $x^{2} + y^{2} - 24 x - 54 y - 88 = 0$
D $x^{2} + y^{2} + 24 x + 36 y + 62 = 0$

Verified Solution

Answer & Solution

Correct Answer

(B) $x^{2} + y^{2} - 12 x - \frac{126}{5} y - \frac{212}{5} = 0$

Step-by-step Solution

Detailed explanation

Given: C1≡x2+y2-4x-6y-12=0 C2≡x2+y2+6x+18y+26=0 Now, we know that length of tangent drawn from any point to circle is given by S1. Now, it is given that, S1S2=23 ⇒S1S2=49 ⇒9S1-4S2=0 ⇒9h2+k2-4h-6k-12-4h2+k2+6h+18k+26=0 ⇒5h2+5k2-60h-126k-212=0…

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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If Px1,y1 is a point such that the lengths of the tangents from it to the circles x2+y2-4x-6y-12=0 and x2+y2+6x+18y+26=0 are in the ratio 2:3, then the locus of P is

Step-by-step Solution

If $P (x_{1}, y_{1})$ is a point such that the lengths of the tangents from it to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ are in the ratio $2 : 3$ , then the locus of $P$ is