Question

Prove that the curves y2=4axandxy=c2 cut at right angles if c4=32a4

Answer 1

Answer:

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Let (x, y) be a point on both curves.  So x and y satisfy:

y² = 4ax          ... (1)

xy = c²            ... (2)

Express the gradients of the tangents in terms of x and y.

First curve: y² = 4ax => 2y × dy/dx = 4a  =>  dy/dx = 2a/y

Second curve: xy = c²  =>  x dy/dx + y = 0  =>  dy/dx = -y/x

The two curves cut at right angles

<=> the tangents are at right angles

<=> 2a/y = -1 / ( -y/x ) = x/y

<=> 2a = x

<=> 8a³ = x³

<=> 32a⁴ = 4ax³ = (4ax) x²

<=> 32a⁴ = y²x²   [ by equation (1) ]

<=> 32a⁴ = (yx)² = (c²)²    [ by equation (2) ]

<=> 32a⁴ = c⁴