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

1

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2

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3

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4

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Solution

The correct option is D $2$

$y = x^{2}; y = - {(x - 2)}^{2}$
$\frac{α^{2} + {(β - 2)}^{2}}{α - β} = 2 α = - 2 (β - 2)$
$\Rightarrow α = 2 - β \Rightarrow β = 2 - α$
$\frac{α^{2} + α^{2}}{α - 2 + α} = 2 a \Rightarrow \frac{2 α^{2}}{2 α - 2} = 2 α$
$α^{2} = α (2 α - 2)$
$\Rightarrow α^{2} = 2 α^{2} - 2 α$
$α^{2} = 2 α \Rightarrow α = 0, 2$
$α = 0; β = 2$
$α = 2; β = 0$
Hence two common tangent

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