The line $y=2x+k$ cuts the circle $(x-2)^2+(y+1)^2=25$ at $A$ and $B$. The length of $AB$

Question

The line $y=2x+k$ cuts the circle $(x-2)^2+(y+1)^2=25$ at $A$ and $B$. The length of $AB$ is $6$ units. Find the two possible values of $k$, giving your answers in exact form.[5] Find the midpoint of $AB$ for each value of $k$.[2]

Accepted Answer

(a) $k=-5\pm4\sqrt5$. (b) $M=\left(2\mp\dfrac{8\sqrt5}{5},-1\pm\dfrac{4\sqrt5}{5}\right)$, with corresponding signs.

An oblique chord of a given length

An oblique chord of a given length

Solution:

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