First slide

Definition of a circle

Question

The point on the straight line $y = 2 x + 11$ which is nearest to the circle 16 $(x^{2} + y^{2}) + 32 x - 8 y - 50 = 0$ , is

Moderate

A

$(9 / 2, 2)$
B

$(- 9 / 2, 2)$
C

$(9 / 2, - 2)$
D

none of these

Solution

Clearly, required point is the point of intersection of the line $y = 2 x + 11$ and the line perpendicular to it passing through the centre of the circle.
The coordinates of the centre are $(- 1, 1 / 4)$
The equation of the line through $(- 1, 1 / 4)$ and perpendicular to $y = 2 x + 11$ is
$y - \frac{1}{4} = - \frac{1}{2} (x + 1) \Rightarrow 2 x + 4 y + 1 = 0$
Clearly, $(- 9 / 2, 2)$ is the point of intersection of $y = 2 x + 11$ and $2 x + 4 y + 1 = 0$ .
So, the coordinates of the required point are $(- 9 / 2, 2)$

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