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Power of a Point

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Question

Let x & y be the real numbers satisfying the equation $x^{2} - 4 x + y^{2} + 3 = 0$ .If the maximum and minimum values of $x^{2} + y^{2}$ are M & m respectively , then find the numerical value of (M+m).

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Solution

$x^{2} - 4 x + y^{2} + 3 = 0$

$\Rightarrow x^{2} - 4 x + 4 + y^{2} - 1 = 0$

${(x - 2)}^{2} + y^{2} = 1$

represents a circle with center at $(2, 0)$ $&$ radius $= 1$ units

$∴ x^{2} + y^{2} \leq 3$

$x^{2} + y^{2} \geq 1$

$∴ M = 3, m = 1$

$∴ M + m = 4$

Hence, the answer is $4.$

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