First slide

Parabola

Question

An equation of a common tangent to the parabola
$y^{2} = 4 x and x^{2} = 4 y$ is

Moderate

A

$x - y + 1 = 0$
B

$x + y - 1 = 0$
C

$x + y + 1 = 0$
D

$y = 0$

Solution

Equation of a tangent to $y^{2} = 4 x is y = m x + 1 / m$
If it touches the parabola $x^{2} = 4 y$ then the roots of
the equation $x^{2} = 4 (m x + 1 / m)$ are coincident. So
$(4 m)^{2} + \frac{16}{m} = 0 \Rightarrow m = - 1$ and the equation of
the tangent is $x + y + 1 = 0 .$

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