First slide

Definition of a circle

Question

The line $3 x - 2 y = k$ meets the circle $x^{2} + y^{2} = 4 r^{2}$ at only one point, if $k^{2} =$

Moderate

A

$20 r^{2}$
B

$52 r^{2}$
C

$\frac{52}{9} r^{2}$
D

$\frac{20}{9} r^{2}$

Solution

The line $3 x - 2 y - k = 0$ meets the circle $x^{2} + y^{2} = 4 r^{2}$ at only one point. So, it is a tangent to the circle.
$∴ |\frac{3 \times 0 - 2 \times 0 - k}{\sqrt{3^{2} + (- 2)^{2}}}| = 2 r \Rightarrow k^{2} = 52 r^{2}$

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