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Equation of common tangent of  y=x^2, y=-x^2+4 x-4 is

Option: 1

y=4(x-1)$ and $y=0


Option: 2

y=0$ and $y=-4(x-1)


Option: 3

y=-4(x-1)$ and $y=-30 x-50


Option: 4

y=-30 x-50$ and $y=4(x-1)


Answers (1)

best_answer

The equation of tangent to   y=x^2, be y=m x-\frac{m^2}{4}

Putting in  y=-x^2+4 x-4  only get one value of x i.e. Discriminant must be zero. 

\begin{aligned} & \therefore m x-\frac{m^2}{4}=-x^2+4 x-4 \\ \Rightarrow & x^2+x(m-4)+4-\frac{m^2}{4}=0 \\ D & =0 \end{aligned}

Now

\begin{aligned} & (m-4)^2\left(16-m^2\right)=0 \\ \Rightarrow & 2 m(m-4)=0 \\ \Rightarrow & m=0,4 \\ \therefore & y=0 \text { and } y=4(x-1) \end{aligned}are the required tangents.

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Riya

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