Question

Find the measure of the angle between y²=4ax and x²=4ay.

Answer 1

first of all, we have to find intersecting points.
y² = 4ax and x² = 4ay

y² = 4ax
y⁴ = 16a²x²
y⁴ = 16a²(4ay)
y⁴ = 64a³y
y = 0, 4a

put , y = 0 in equation y² = 4ax => x = 0
put , y = 4a in equation y² = 4ax => x = 4a

hence, there are two intersecting points e.g., (0,0) and (4a,4a)

now find slope of tangents of curve :
y² = 4ax
differentiate with respect to x,
2y.dy/dx = 4a
dy/dx = 2a/y
at (0,0) , slope of tangent = 2a/0 = ∞
at (4a,4a), slope of tangent = 2a/4a = 1/2

x² = 4ay
differentiate with respect to x,
2x = 4a.dy/dx
dy/dx = x/2a
at (0,0),slope of tangent = 0/2a = 0
at (4a, 4a), slope of tangent = 4a/2a = 2

now, angle between curves at (0,0)
= tan^-1{|0 - ∞|}/{1 +0.∞}
= tan^-1{∞}
= 90°

angle between curves at (4a,4a)
= tan^-1{2 - 1/2}/{1 + 2.1/2}
= tan^-1{4 - 1}/{2(1 + 1)}
= tan^-1 (3/4)
≈ 37°

[note :- angle between two curves = tan^-1|(m2-m1)/(1+m1.m2)| , where m1 and m2 are slope of tangents of curves]

Answer 2

Dear Student:

Clearly (0,0) is one point.

And angle is π/2

Because slope of first curve at 0 is 0

and second curve is ∞

Use formula of tangent angle.

Se the attachment.