First slide

Hyperbola in conic sections

Question

Equation of a common tangent to the curves $y^{2} = 8 x$ and $x y = - 1$ is

Moderate

A

$3 y = 9 x + 2$
B

$y = 2 x + 1$
C

$2 y = x + 8$
D

$y = x + 2$

Solution

Equation of a tangent at $(a t^{2}, 2 a t)$ to $y^{2} = 8 x$
$t y = x + a t^{2}$ where $4 a = 8 i.e. a = 2$
$\Rightarrow t y = x + 2 t^{2}$ which intersects the curve $x y = - 1$
at the points given by $\frac{x (x + 2 t^{2})}{t} = - 1$ clearly $t \neq 0$
or $x^{2} + 2 t^{2} x + t = 0$ and will be a tangent to the curve if the roots of this quadratic equation are equal, for which
$4 t^{4} - 4 t = 0 \Rightarrow t = 0$ or t = 1 and an equation of a common tangent is $y = x + 2 .$

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