Question

y=x²,x²+y²=20,Find the measure of the angle between curves,if they intersect.

Answer 1

first of all, we have to find intersecting points.

y = x² and x² + y² = 20

x² + y² = 20
y + y² = 20
y² + y - 20 = 0
y² + 5y - 4y - 20 = 0
y(y + 5) - 4(y + 5) = 0
(y - 4)(y + 5) = 0
y = 4 and -5

put y = 4 , in equation y = x² => x = ±4
put y = -5 , in equation y = x² => -5 ≠ x²

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

slope of tangents of curves :-
y = x² ,
differentiate with respect to x,
dy/dx = 2x
at (4, 4) , slope of tangent = 2 × 4 = 8
at (-4, 4), slope of tangent = 2 × -4 = -8

x² + y² = 20
differentiate with respect to x,
2x + 2y.dy/dx = 0
dy/dx = -x/y
at (4,4) , slope of tangent = -4/4 = -1
at (-4,4), slope of tangent = -(-4)/4 = 1

angle between curves at (4,4)
= tan^-1|{8 - (-1)}/{1 + 8 × (-1)}|
= tan^-1 {9/7}

angle between curves at (-4, 4)
= tan^-1|{-8-1}{1 + (-8)(1)}|
= tan^-1(9/7)

hence, angle between curves is tan^-1 (9/17)

Answer 2

Dear student:

y=x²           ................(1)

x²+y²=20         ...............(2)

Find slope of (1) and (2)

and see the intersection of both curve.

We got point of intersection(2,4) and (-2,4)

Acute angle at both point is same.

See the attachment.