The equations of the common tangents to the parabola y = x 2 and y = - ( x - 2 ) 2 is/are [2006]

Question

The equations of the common tangents to the parabola $y = x^{2}$ and $y = - {(x - 2)}^{2}$ is/are [2006]

Smart Achievers · Accepted Answer

(1, 2)

If $y = m x + c$ is tangent to $y = x^{2}$ then $x^{2} - m x - c = 0 has equal roots$

$\Rightarrow m^{2} + 4 c = 0 \Rightarrow c = - \frac{m^{2}}{4}$

$∴$ $y = m x - \frac{m^{2}}{4} is tangent to y = x^{2}$

$∴$ $This is also tangent to y = - {(x - 2)}^{2}$

$\Rightarrow m x - \frac{m^{2}}{4} = - x^{2} + 4 x - 4$

$\Rightarrow x^{2} + (m - 4) x + (4 - \frac{m^{2}}{4}) = 0 has equal roots$

$∴$ $m^{2} - 8 m + 16 = - m^{2} + 16 \Rightarrow m = 0, 4$

$∴$ $y = 0 or y = 4 x - 4 are the tangents.$

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