If y=mx+c is a common tangent to the parabola | vedclass

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(C) The equation of a tangent to $y^2=4\sqrt{k}x$ is $y=mx+\frac{\sqrt{k}}{m}$,where $m$ is the slope of the tangent.If it touches the circle $x^2+y^2

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(C) The equation of a tangent to $y^2=4\sqrt{k}x$ is $y=mx+\frac{\sqrt{k}}{m}$,where $m$ is the slope of the tangent.If it touches the circle $x^2+y^2=\frac{k}{2}$,the perpendicular distance from the center $(0,0)$ to the line $mx-y+\frac{\sqrt{k}}{m}=0$ must equal the radius $\sqrt{\frac{k}{2}}$.$\left|\frac{\sqrt{k}/m}{\sqrt{m^2+1}}ight| = \sqrt{\frac{k}{2}}$$\frac{k}{m^2(m^2+1)} = \frac{k}{2}$$m^2(m^2+1) = 2$$m^4+m^2-2 = 0$$(m^2+2)(m^2-1) = 0$Since $m$ must be real,$m^2=1$,which gives $m=1$ or $m=-1$.The product of the slopes is $m_1 	imes m_2 = 1 	imes (-1) = -1$.

If $y=mx+c$ is a common tangent to the parabola $y^2=4\sqrt{k}x$ and the circle $2x^2+2y^2=k$,then the product of the slopes of such common tangents is

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