MHT CET · Maths · Circle

If the sides of a rectangle are given by the equations \(x=-2, x=6, y=-2, y=5\), then the equation of the circle, drawn on the diagonal of this rectangle as its diameter, is

A \(x^2+y^2+4 x+3 y+22=0\)
B \(x^2+y^2-4 x+3 y-22=0\)
C \(x^2+y^2-4 x-3 y-22=0\)
D \(x^2+y^2+4 x-3 y+22=0\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(x^2+y^2-4 x-3 y-22=0\)

Step-by-step Solution

Detailed explanation

The given equations of the sides are \(x=-2, x=6, y=-2\) and \(y=5\).

Here, the diagonals AC and BD of rectangle ABCD are diameters of the circle passing through the vertices \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D .
Considering diagonal AC with end points \(A(-2,-2)\) and \(C(6,5)\), we get
Equation of circle in diameter form as,
\(\begin{aligned}
& (x-6)(x+2)+(y-5)(y+2)=0 \\
& \Rightarrow x^2-4 x-12+y^2-3 y-10=0 \\
& \Rightarrow x^2+y^2-4 x-3 y-22=0
\end{aligned}\)

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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