AP EAMCET · Maths · Circle

If a line drawn from a fixed point $M (a, b)$ cuts the circle $x^{2} + y^{2} = k^{2}$ at $C$ and $D,$ then $M C \times M D$ is equal to

A $a^{2} + b^{2} + k^{2}$
B $a^{2} + b^{2} - k^{2}$
C $a^{2} - b^{2} - k^{2}$
D $x^{2}$

Verified Solution

Answer & Solution

Correct Answer

(B) $a^{2} + b^{2} - k^{2}$

Step-by-step Solution

Detailed explanation

Let the line have slope tanθ x-acosθ=y-bsinθ=r x=a+rcosθ, y=b+rsinθ ⇒x2+y2=k2 ⇒a+rcosθ2+(b+rsinθ)2=k2 r2+2r(acosθ+bsinθ)+a2+b2-k2=0 It has two roots r1=MC and r2=MD r1·r2=a2+b2-k2 MC×MD=a2+b2-k2

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.

Same subject

Explore more questions on app

From AP EAMCET

Explore more questions on app

If a line drawn from a fixed point Ma, b cuts the circle x2+y2=k2 at C and D, then MC×MD is equal to

Step-by-step Solution

If a line drawn from a fixed point $M (a, b)$ cuts the circle $x^{2} + y^{2} = k^{2}$ at $C$ and $D,$ then $M C \times M D$ is equal to