First slide

Definition of a circle

Question

Let x and y be real variables satisfying $x^{2} + y^{2} + 8 x - 10 y - 40 = 0$ .Let $a = max \{\sqrt{(x + 2)^{2} + (y - 3)^{2}}\}$ and $b = min \{\sqrt{(x + 2)^{2} + (y - 3)^{2}}\}$ . Then,

Moderate

A

$a + b = 18$
B

$a + b = \sqrt{2}$
C

$a - b = 4 \sqrt{2}$
D

$a \cdot b = 73$

Solution

$x^{2} + y^{2} + 8 x - 10 y - 40 = 0$
The center of the circle is (4, 5).
its radius is 9.
Distance of the center (-4,5) from the point (-2,3) is $\sqrt{4 + 4} = 2 \sqrt{2}$ .
Therefore,
$\begin{array}{ll} a = 2 \sqrt{2} + 9 and b = - 2 \sqrt{2} + 9 \\ ∴ a + b = 18 \\ a - b = 4 \sqrt{2} \\ a \cdot b = 81 - 8 = 73 \end{array}$

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