Question

xy=6, x²y=12,Find the measure of the angle between curves,if they intersect.

Answer 1

first of all, we have to find intersecting points.
xy = 6 and x²y = 12

x²y = 12
=> x(xy) = 12
=> x × 6 = 12
=> x = 2
now, put x = 2 in equation xy = 6 => y = 3

so, there is only one intersecting point of given curves e.g., (2,3)
now, we have to find out slope of tangents of curves at (2,3)

xy = 6
differentiate with respect to x,
x.dy + y.dx = 0
dy/dx = -y/x
at (2, 3), slope of tangent, m1 = -3/2

x²y = 12
differentiate with respect to x,
2xy + x²dy/dx = 0
dy/dx = -2y/x
at (2,3) , slope of tangent , m2 = -2 × 3/2 = -3

now, angle between curves = tan^-1|(m2-m1)/(1+m1.m2)|
= tan^-1|(-3 + 3/2)/(1 + (-3)(-3/2)|
= tan^-1|(-3/2)/(2 + 9)/2|
= tan^-1 (3/11)

Answer 2

Dear student:

xy=6           ................(1)

x²y=12         ...............(2)

Find slope of (1) and (2)

and see the intersection of both curve.

See the attachment.