The number of common tangents to the parabola | vedclass

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(A) Given parabolas are $y = x^{2}$ and $y = -(x-2)^{2}$.Let the tangent to $y = x^{2}$ be $y = mx - \frac{m^{2}}{4}$.This line is also a tangent to $

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$1$

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(A) Given parabolas are $y = x^{2}$ and $y = -(x-2)^{2}$.Let the tangent to $y = x^{2}$ be $y = mx - \frac{m^{2}}{4}$.This line is also a tangent to $y = -(x-2)^{2}$, which can be written as $(y-0) = -1(x-2)^{2}$.The condition for a line $y = mx + c$ to be a tangent to $y = a(x-h)^{2} + k$ is $c = k - \frac{m^{2}}{4a}$.Here, $a = -1, h = 2, k = 0$. So, $c = 0 - \frac{m^{2}}{4(-1)} = \frac{m^{2}}{4}$.Equating the two expressions for $c$:$-\frac{m^{2}}{4} = \frac{m^{2}}{4}$ $\Rightarrow \frac{m^{2}}{2} = 0$ $\Rightarrow m = 0$.For $m = 0$, the tangent is $y = 0$.Since there is only one value of $m$, there is only $1$ common tangent.

The number of common tangents to the parabolas $y = x^{2}$ and $y = -x^{2} + 4x - 4$ is:

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