• # question_answer The equation of the common tangent to the curves ${{y}^{2}}=8x$ and $\mathbf{xy}\text{=}-\mathbf{1}$is- A)  $2y=x+8$                   B)  $y=x+2$      C)  $y=2x+1$       D)  $3y=9x+2$

[b] Equation of the tangent of the curve ${{y}^{2}}=8x\,be\,y=mx+\frac{2}{m}$ And it must satisfy the equation $-xy=-1$ $\Rightarrow x\left( mx+\frac{2}{m} \right)=-1\Rightarrow m{{x}^{2}}+\frac{2}{m}x+1=0$ Which is quadratic equation in x and its roots be equal. i.e. D=0 $\Rightarrow {{b}^{2}}-4ac=0{{\left( \frac{2}{m} \right)}^{2}}-4.m.1=0$ $\Rightarrow {{m}^{3}}=1$ $\therefore m=1$ Hence, equation of the comment tangent be $y=x+2$ Hence option [b] is correct.