• question_answer Direction: A straight line will touch a given conic if there is only one point of intersection of the line and the given conic. If the conic is specified by quadratic equation in$x$ and $y,$ then the straight line will touch if the discriminant of the equation obtained by the elimination of one of the variable is zero. Let us consider parabola ${{y}^{2}}=8x$ and an ellipse$15{{x}^{2}}+4{{y}^{2}}=60$. The equation of a tangent common to both the parabola and the ellipse is A)  $2x-y-8=0$   B)  $x-2y+8=0$ C)  $2x-y+8=0$ D)  $x+2y-8=0$

Let the tangent of ${{y}^{2}}=8x$ is $y=mx+2\,/m$ and the tangent of $\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{15}=1$ is $y=mx\,\pm \,\sqrt{4{{m}^{2}}+15}$ From Eqs. (i) and (ii), we get $\frac{2}{m}=\pm \,\sqrt{4{{m}^{2}}+15}$ $\Rightarrow$            $4{{m}^{4}}+15{{m}^{2}}-4=0$ $\Rightarrow$            $m=\pm \,\frac{1}{2}$ Hence, equation of tangent is $atm=\frac{1}{2}$ $\Rightarrow$            $y=\frac{x}{2}+4$ $\Rightarrow$            $2y=x+8$
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