Angle Between Two Curves, 3

Category: Differential Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the angle of intersection between the two curves:

Solution:

Consider the given pair of two curves above 

  

The first thing that we need to do is to get their point of intersection by solving the systems of equation as follows

Substitute this value of y to the second given equation, we have

The value of y is

The point of intersection of the given two curves is P(0, 1). 

Consider the first given equation of a curve 

Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have

The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.

To get the value of the slope of a curve at their point of intersection, substitute x = 0 and y = 1 at the equation above, we have

The slope of a curve at their point of intersection is equal to the slope of tangent line that passes thru also at their point of intersection. Hence, 

Consider the second given equation of a curve

Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have

The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.

To get the value of the slope of a curve at their point of intersection, substitute x = 0 and y = 1 at the equation above, we have
 

 

 

 

   
The slope of a curve at their point of intersection is equal to the slope of tangent line that passes thru also at their point of intersection. Hence, 

Therefore, the angle between two curves at their point of intersection is

or

 

Angle Between Two Curves, 2

Category: Differential Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the angle of intersection between the two curves:

Solution:

Consider the given pair of two curves above

The first thing that we need to do is to get their point of intersection by solving the systems of equation as follows

but

Hence, the above equation becomes

The value of y is

The point of intersection of the given two curves is P(1, 2).

Consider the first given equation of a curve

Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have 

The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve. Actually, the first curve is a straight line. 

Since the right side of the equation contains coefficient only, the slope of the first curve is 

Consider the second given equation of a curve 

Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have 

The slope of a curve is equal to the first derivative of the equation of a curve with respect to x. In this case, dy/dx is the slope of a curve.  

To get the value of the slope of a curve at their point of intersection, substitute x = 1 and y = 2 at the equation above, we have 

The slope of a curve at their point of intersection is equal to the slope of tangent line that passes thru also at their point of intersection. Hence,  

Therefore, the angle between two curves at their point of intersection is 

or