Question

The equation of the common tangent of the parabolas
$x2 = 108y$
$x2 = 108yandy2 = 32$
​

Answer 1

SOLUTION

TO DETERMINE

The equation of the common tangent to the parabolas

$\sf{ {x}^{2} = 108y \: \: \: and \: \: {y}^{2} = 32x }$

EVALUATION

Here the given equation of the parabolas are

$\sf{ {x}^{2} = 108y \: - - - - (1) }$

Comparing with the general equation x² = 4by we get

4b = 108

⇒ b = 27

$\sf{ {y}^{2} = 32x \: \: \: - - - - (2) }$

Comparing with the general equation y² = 4ax we get

4a = 32

⇒ a = 8

We know that the equation of the common tangent to the parabolas y² = 4ax and x² = 4by is

$\sf{ \sqrt[3]{b} \: y + \sqrt[3]{a} \: x + \sqrt[3]{ {a}^{2} {b}^{2} } = 0}$

Putting the values of a and b we get

$\sf{ \sqrt[3]{27} \: y + \sqrt[3]{8} \: x + \sqrt[3]{ {8}^{2} \times {27}^{2} } = 0}$

$\sf{ \implies \: 3y + 2 x + (4 \times 9) = 0}$

$\sf{ \implies \: 3y + 2 x + 36 = 0}$

$\sf{ \implies \: 2 x + 3y + 36 = 0}$

FINAL ANSWER

The equation of the common tangent

$\boxed{ \: \: \sf{ 2 x + 3y + 36 = 0} \: \: }$

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