TS EAMCET · Maths · Circle

For the circle \(x-2=5 \cos \theta, y+1=5 \sin \theta\) where \(\theta\) is the parameter, the line \(x=1+\frac{\mathrm{r}}{2}\), \(y=-2+\frac{\sqrt{3}}{2} \mathrm{r}\) where r is the parameter, is a

A Chord of the circle other than diameter
B Tangent of the circle
C Diameter of the circle
D Line that does not meet the circle

Verified Solution

Answer & Solution

Correct Answer

(A) Chord of the circle other than diameter

Step-by-step Solution

Detailed explanation

Circle: \((x-2)^2 + (y+1)^2 = 5^2\) Center \(C=(2, -1)\), Radius \(R=5\) Line: \(y+2 = \sqrt{3}(x-1)\) \(\sqrt{3}x - y - (2+\sqrt{3}) = 0\) Distance \(d = \frac{|\sqrt{3}(2) - (-1) - (2+\sqrt{3})|}{\sqrt{(\sqrt{3})^2 + (-1)^2}}\)…

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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