MHT CET · Maths · Circle

The abscissae of two points \(\mathrm{A}\) and \(\mathrm{B}\) are the roots of the equation \(x^2+2 a x-b^2=0\) and their ordinates are roots of the equation \(y^2+2 \mathrm{p} y-\mathrm{q}^2=0\). Then the equation of the circle with \(\mathrm{AB}\) as diameter is given by

A \(x^2+y^2-2 \mathrm{a} x-2 \mathrm{p} y+\left(\mathrm{b}^2+\mathrm{q}^2\right)=0\)
B \(x^2+y^2-2 \mathrm{a} x-2 \mathrm{p} y-\left(\mathrm{b}^2+\mathrm{q}^2\right)=0\)
C \(x^2+y^2+2 \mathrm{a} x+2 \mathrm{p} y+\left(\mathrm{b}^2+\mathrm{q}^2\right)=0\)
D \(x^2+y^2+2 \mathrm{a} x+2 \mathrm{p} y-\left(\mathrm{b}^2+\mathrm{q}^2\right)=0\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(x^2+y^2+2 \mathrm{a} x+2 \mathrm{p} y-\left(\mathrm{b}^2+\mathrm{q}^2\right)=0\)

Step-by-step Solution

Detailed explanation

Let \(\mathrm{A} \equiv\left(x_1, y_1\right)\) and \(\mathrm{B} \equiv\left(x_2, y_2\right)\).
According to the given condition,
\(x_1+x_2=-2 \mathrm{a}, x_1 x_2=-\mathrm{b}^2\)
\(y_1+y_2=-2 \mathrm{p}, y_1 y_2=-\mathrm{q}^2\)
The equation of the circle with \(\mathrm{A}\left(x_1, y_1\right)\) and \(\mathrm{B}\left(x_2, y_2\right)\) as the end points of diameter is
\(\left(x-x_1\right)\left(x-x_2\right)+\left(y-y_1\right)\left(y-y_2\right)=0\)
\(\Rightarrow x^2+y^2-x\left(x_1+x_2\right)-y\left(y_1+y_2\right)+x_1 x_2~+\) \(y_1 y_2=0\)
\(\Rightarrow x^2+y^2+2 \mathrm{a} x+2 \mathrm{p} y-\mathrm{b}^2-\mathrm{q}^2=0\)

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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