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A circle passing through the point $P (α, β)$ in the first quadrant touches the two coordinate axes at the point A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of $αβ$ is _____

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answer is 121.

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Detailed Solution

Let the circle is $x^{2} + y^{2} - 2 rx - 2 ry + r^{2} = 0$

Given, $P (α, β)$ lies on circle
$\Rightarrow α^{2} + β^{2} - 2 r α - 2 rβ + r^{2} = 0$

Equation of line AB is x + y = r

Given perpendicular distance from $P (α, β) to x + y = r is 11$

$\Rightarrow \frac{| α + β - r |}{\sqrt{2}} = 11$

$| α + β - r | = 11 \sqrt{2}$

S.O.B.S

$α^{2} + β^{2} + r^{2} + 2 αβ - 2 β r - 2 α r = 121 \times 2$

$\Rightarrow 2 αβ = 121 \times 2$

$αβ = 121$

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