Prove that the intersection of the curves y² = 4ax and x² = 4by is given by .

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

y² = 4ax and x² = 4by

Put in x² = 4by

⇒ y³ = 64a²b

Put this in x² = 4by

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

Now angle between curves or lines is given by where m₁ and m₂ 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 y² = 4ax

Differentiating with respect to x

Slope at

Calculating slope for x² = 4by

Differentiating with respect to x

Slope at

Put values of m₁ and m₂ from (a) and (b) respectively in

Hence proved