If the angle between the circles x2+y2-4x-6y+k=0 and x2+y2+8x-4y+11=0 is π3, then a value of k is

TS EAMCET · Maths · Circle

If the angle between the circles $x^{2} + y^{2} - 4 x - 6 y + k = 0$ and $x^{2} + y^{2} + 8 x - 4 y + 11 = 0$ is $\frac{π}{3},$ then a value of $k$ is

A $- 36$
B $36$
C $3$
D $2$

Verified Solution

Answer & Solution

Correct Answer

(A) $- 36$

Step-by-step Solution

Detailed explanation

We have S1: x2+y2-4x-6y+k=0 with centre C12, 3 and radius r1=-22+-32-k=13-k units ⇒13-k>0⇒k<13 ...(1); S2: x2+y2+8x-4y+11=0 with centre C2-4, 2 and radius r2=42+-22-11=3 units. Now, the angle between the circles is given by…

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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