Question

y²=8x,x²=27y,Find the measure of the angle between curves,if they intersect.

Answer 1

first of all, we have to find intersecting points
y² = 8x and x² = 27y

y² = 8x
taking square both sides ,
y⁴ = 64x² => y⁴ = 64 × 27y
y⁴ = 64 × 27y
y(y³ - 64 × 27) = 0
y = 0, 12
put y = 0, in y² = 8x => x = 0
put y = 12 in y² = 8x => x = 18

hence, there are two intersecting points (0,0) and (18,12).

find slope of tangent of curves :
y² = 8x
differentiate with respect to x,
2y.dy/dx = 8
dy/dx = 4/y
at (0,0) , slope of tangent = 4/0 = ∞
at (18,12) , slope of tangent = 4/12= 1/3

x² = 27y
2x = 27dy/dx
dy/dx = 2x/27
at (0,0) , slope of tangent = 2 × 0/27 = 0
at (18,12), slope of tangent = 2 × 18/27 = 4/3

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

angle between curves at (18,12)
= tan^-1 {|4/3 - 1/3|}/{1+ 4/3.1/3}
= tan^-1 {2 × 3}/{9 + 4}
= tan^-1 (6/13)

hence, angle between curves : 90° and tan^-1 (6/13)

Answer 2

Dear Student:

Clearly (0,0) is one point.

And angle is π/2

Because slope of first curve is ∞

and second curve is 0

Use formula of tangent angle.

See the attachment.