Number of common tangents of $y = x^{2}$ and $y = - x^{2} + 4 x - 4$ is

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Solution

The correct option is C 2
$y = x^{2}; y = - x^{2} + 4 x - 4$

Let the commom tangent be,
$y = m x + c$
Finding the intersection with the parabola $y = x^{2}$
$x^{2} = m x + c \Rightarrow x^{2} - m x - c = 0$
Condition for tangency,
$D = 0 \Rightarrow m^{2} + 4 c = 0 \dots (1)$
Finding the intersection with the parabola $y = - x^{2} + 4 x - 4$ ,
$- x^{2} + 4 x - 4 = m x + c \Rightarrow x^{2} + x (m - 4) + (4 + c) = 0$
Condition for tangency,
$D = 0 \Rightarrow (m - 4)^{2} - 4 (4 + c) = 0 \Rightarrow m^{2} - 8 m + 16 - 16 - 4 c = 0$
Using (1),
$\Rightarrow 2 m^{2} - 8 m = 0 \Rightarrow m = 0, 4$

Hence, there are two common tangents to both the curves.

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