TS EAMCET · Maths · Circle

Let $P$ be any point on the circle $x^{2} + y^{2} - 2 x - 1 = 0$ and $C$ be its centre. Let $A B$ be the chord of contact of $P$ with respect to the circle $x^{2} + y^{2} - 2 x = 0$ . Then the locus of the circumcentre of the triangle $C A B$ is

A $2 x^{2} + 2 y^{2} - 4 x + 1 = 0$
B $x^{2} + y^{2} - 4 x + 2 = 0$
C $x^{2} + y^{2} - 4 x + 1 = 0$
D $2 x^{2} + 2 y^{2} - 4 x + 3 = 0$

Verified Solution

Answer & Solution

Correct Answer

(A) $2 x^{2} + 2 y^{2} - 4 x + 1 = 0$

Step-by-step Solution

Detailed explanation

Plotting the diagram of the given data we have, Now let the circle S1≡x2+y2-2x-1=0 whose radius is 2 and centre C1,0, And circle S2≡x2+y2-2x=0 whose radius is 1 and centre C1,0 now we can see that radius of S1 is 2 times of radius of S2, so S1 will be director circle…

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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Let P be any point on the circle x2+y2-2x-1=0 and C be its centre. Let AB be the chord of contact of P with respect to the circle x2+y2-2x=0. Then the locus of the circumcentre of the triangle CAB is

Step-by-step Solution

Let $P$ be any point on the circle $x^{2} + y^{2} - 2 x - 1 = 0$ and $C$ be its centre. Let $A B$ be the chord of contact of $P$ with respect to the circle $x^{2} + y^{2} - 2 x = 0$ . Then the locus of the circumcentre of the triangle $C A B$ is