Question

Find the measure of the angle between y=x²,y=(x-2)²

Answer 1

In the attachments I have answered  this problem.       Concept:    The angle between two curve is the angle between their tangents at the  point of intersection.     Slope of tangents of the given curve is calculated by derivative method.       See the attachment for detailed solution.

Answer 2

angle between two curve : find slope of both curves . Let $m_1$ and $m_2$ are the slopes of given curves.
then, angle between them =$\tan^{-1}\frac{|m_2-m_1|}{|1+m_1.m_2|}$

first of all we have to find intersecting point of two given curves.
y = x² and y = (x - 2)²
=> x² = (x - 2)²
=> x² = x² - 4x + 4
=> x = 1

at x = 1 , y = 1
so, intersecting point is (1,1)

now find slope of tangents of both curves .
slope of tangent of y = x² :
differentiate y = x² with respect to x,
dy/dx = 2x
at (1,1) slope of tangent of 1st curve = 2(1) = 2
e.g., $m_1$= 2

slope of tangent of y = (x - 2)² :
differentiate y = (x - 2)² with respect to x,
dy/dx = 2(x - 2)
at(1,1) slope of tangent of 2nd curve = 2(1-2)=-2
e.g., $m_2$ = -2

now , angle between them = $tan^{-1}\frac{|-2-2|}{1+(-2)(2)|}$
= $tan^{-1}\frac{4}{3}$
≈ 53°