Let the equation x(x+2)(12-k) = 2 have equal roots.

Question

Let the equation $x(x+2)(12-k) = 2$ have equal roots. Then the distance of the point $\left(k,\dfrac k2\right)$ from the line $3x+4y+5=0$ is

Accepted Answer

Rewrite the equation (for $k \ne 12$) as
$$x^2+2x - \frac{2}{12-k} = 0$$

For equal roots, $D=0$:
$$4 + \frac{8}{12-k} = 0 \implies 8 = -4(12-k) \implies 8=-48+4k \implies k=14$$

The point is $(14,7)$. Distance from $3x+4y+5=0$:
$$d = \frac{|3(14)+4(7)+5|}{\sqrt{3^2+4^2}} = \frac{|42+28+5|}{5} = \frac{75}{5} = 15$$

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