TS EAMCET · Maths · Straight Lines

From a point $A (0, 3)$ on the circle ${(x + 2)}^{2} + {(y - 3)}^{2} = 4$ , a chord $A B$ is drawn and it is extended to a point $Q$ such that $A Q = 2 A B$ . Then the locus of $Q$ is

A ${(x + 4)}^{2} + {(y - 3)}^{2} = 16$
B ${(x + 1)}^{2} + {(y - 3)}^{2} = 32$
C ${(x + 1)}^{2} + {(y - 3)}^{2} = 4$
D ${(x + 1)}^{2} + {(y - 3)}^{2} = 1$

Verified Solution

Answer & Solution

Correct Answer

(A) ${(x + 4)}^{2} + {(y - 3)}^{2} = 16$

Step-by-step Solution

Detailed explanation

Given, From a point A0,3 on the circle x+22+y-32=4, a chord AB is drawn and it is extended to a point Qh,k such that AQ=2 AB. Which means Bx,y is the midpoint of A0,3 & Qh,k Now using the midpoint formula we get, x=h2 & y=k+32 Now putting the…

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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From a point A0,3 on the circle x+22+y-32=4, a chord AB is drawn and it is extended to a point Q such that AQ=2 AB. Then the locus of Q is

Step-by-step Solution

From a point $A (0, 3)$ on the circle ${(x + 2)}^{2} + {(y - 3)}^{2} = 4$ , a chord $A B$ is drawn and it is extended to a point $Q$ such that $A Q = 2 A B$ . Then the locus of $Q$ is