Equation of common tangent to the curves y^2 = 8x and xy = -1 is

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10-2. Parabola, Ellipse, Hyperbola

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The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is

$3y = 9x +2$

$y = 2x +1$

$2y = x+8$

$y = x +2$

Solution

Any point on $y^2 = 8x$ is $(2t_2 , 4t)$ where the tangent is $yt = x + 2t_2$.Solving it with $xy = -1, y(yt – 2t_2 ) = -1$ or $ty_2 – 2t_2y +1 = 0$ . For common tangent, it should have equal roots.

$\therefore 4t^4 – 4t = 0 \Rightarrow t = 0, 1$ .

$\therefore$ The common tangent is $y = x +2$ , (whent $t= 0$ , it is $x = 0$ which can touch $xy = -1$ atinfinity only).

Standard 11

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