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General Solution of a Differential Equation

Let x y be ...

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

$x^{2} + y^{2} - 4 x + 3 = 0$
center $(2, 0) r = 1$
$x = 2 + c o s t y = s i n t$
$x^{2} + y^{2} = 4 + c o s t + 4 c o s t + s i n^{2} t$
$= 5 + 4 c o s t$
$g = x^{2} + y^{2} = 5 + 4 c o s t$
$\frac{d y}{d t} = - 4 s i n t = 0$
$t = 0, π$
$M a x = 5 + 4 c o s (0) = 9.$
$m i n = 5 + 4 c o s (π) = 1$
Sum $= 10$

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