The common tangent of the two parabolas $y^{2} = 4 x$ and $x^{2} = 32 y$ meets the coordinate axes at $A, B$ respectively. The equation of the circumcircle of $Δ O A B$ is

x^{2} + y^{2} - 4 x - 2 y = 0

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x^{2} + y^{2} + 4 x + 2 y = 0

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x^{2} + y^{2} + 2 x + y = 0

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x^{2} + y^{2} - 2 x - y = 0

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Solution

The correct option is B $x^{2} + y^{2} + 4 x + 2 y = 0$
Equation of a tangent to $y^{2} = 4 x$ in slope form is
$y = m x + \frac{1}{m} (∵ a = 1)$
It also touches $x^{2} = 32 y$
$∴ x^{2} = 32 (m x + \frac{1}{m})$ has equal roots.
$\Rightarrow x^{2} - 32 m x - \frac{32}{m} = 0$ has equal roots.
$\Rightarrow D = 0 \Rightarrow (- 32 m)^{2} + 4 (\frac{32}{m}) = 0$
$\Rightarrow m^{3} = - \frac{1}{8}$
$\Rightarrow m = - \frac{1}{2}$

$∴$ Equation of common tangent is $x + 2 y + 4 = 0$
Here, $A = (- 4, 0), B = (0, - 2)$

Equation of circumcircle of $Δ O A B$ is
$(x + 4) (x - 0) + (y - 0) (y + 2) = 0 (A B is diameter)$
$\Rightarrow x^{2} + y^{2} + 4 x + 2 y = 0$

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The common tangent of the two parabolas y2=4x and x2=32y meets the coordinate axes at A,B respectively. The equation of the circumcircle of ΔOAB is

The common tangent of the two parabolas $y^{2} = 4 x$ and $x^{2} = 32 y$ meets the coordinate axes at $A, B$ respectively. The equation of the circumcircle of $Δ O A B$ is