Answer :

We have to find the angle between parabolas at point of intersection

Angle between parabolas at a point means angle between tangents at those point

Let us first find the point of intersection

y^{2} = 4ax and x^{2} = 4by

Put in x^{2} = 4by

⇒ y^{3} = 64a^{2}b

Put this in x^{2} = 4by

Hence from (i) and (ii) the intersection point is

Now angle between curves or lines is given by where m_{1} and m_{2} are slopes of tangent and θ is required angle between curves

gives us the slope of tangent

Let us find slopes at for both the parabolas

Calculating slope for y^{2} = 4ax

Differentiating with respect to x

Slope at

Calculating slope for x^{2} = 4by

Differentiating with respect to x

Slope at

Put values of m_{1} and m_{2} from (a) and (b) respectively in

Hence proved

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