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Question: The condition that straight line with slope m will be normal to the parabola y<sup>2</sup> = 4ax as ...

The condition that straight line with slope m will be normal to the parabola y2 = 4ax as well as a tangent to the rectangular hyperbola x2 - y2 = a2 is

A

m6 - 4m2 + 2m - 1 = 0

B

m4 + 3m3+2m + 1 = 0

C

m6 - 2m = 0

D

m6 + 4m4 + 3m2 + 1 = 0

Answer

m6 + 4m4 + 3m2 + 1 = 0

Explanation

Solution

The equation of normal to the parabola

y = mx – 2am – am3 if ‘m’ in the slope of the normal if it is a tangent to the hyperbola x2 – y2 =a2 then

(-2am – am3)2 = a2m2 - a2

(2m+m3)2 = m2 – 1

m6 + 4m4 + 4m2 = m2 – 1

m6 + 4m4 + 3m2 + 1 = 0.