જો સમબાજુ ત્રિકોણનું મધ્યકેન્દ્ર \((1, 1)\) હોય અને ત્રિકોણની એક બાજુનું સમીકરણ \(3x + 4y + 3\,= 0\) હોય તો સમબાજુ ત્રિકોણના પરિવર્તુળનું સમીકરણ મેળવો.

A \({x^2} + {y^2} - 2x - 2y - 14 = 0\)
B \({x^2} + {y^2} - 2x - 2y - 2 = 0\)
C \({x^2} + {y^2} - 2x - 2y +2 = 0\)
D \({x^2} + {y^2} - 2x - 2y - 7 = 0\)

Verified Solution

Answer & Solution

Correct Answer

(A) \({x^2} + {y^2} - 2x - 2y - 14 = 0\)

Step-by-step Solution

Detailed explanation

Let radius of circumircles be According to the question, \(\frac{r}{2} = \frac{{10}}{5} \Rightarrow r = 4\) So equation of Required circle is \({\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = 16\) \( \Rightarrow {x^2} + {y^2} - 2x - 2y - 14 = 0\)

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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