Advertisement Remove all ads

Find the acute angle between the curves at their points of intersection, y = x2, y = x3. - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

Find the acute angle between the curves at their points of intersection, y = x2, y = x3.

Advertisement Remove all ads

Solution

The angle between the curves is the same as the angle between their tangents at the points of intersection.
We find the points of intersection of y = x2   ....(1) and  y = x3         .....(2)

From (1) and (2)

x3 = x2 

∴ x3 - x2 = 0

∴ x2(x - 1) = 0

∴ x = 0 or x = 1

When x = 0, y = 0.

When x = 1, y = 1.

∴ the points of intersection are

O = (0, 0) and P = (1, 1)

For y = x2, `"dy"/"dx" = 2"x"`

For y = x3, `"dy"/"dx" = 3"x"^2`

Angle at O = (0, 0)

Slope of tangent to y = x2 at O

`= ("dy"/"dx")_("at O" (0,0)) = 2xx0 = 0`

∴ equation of tangent to y = x2 at O is y = 0.

Slope of tangent to y = x3 at O = `("dy"/"dx")_("at O" (0,0)) = 3 xx 0 = 0`

∴ equation of tangent to y = x3 at P is y = 0.

∴ the tangents to both curves at (0, 0) are y = 0

∴ angle between them is 0.

Angle at P = (1, 1)

Slope of tangent to y = x2 at P

`= ("dy"/"dx")_("at O" (1,1)) = 2xx1 = 2`

∴ equation of tangent to y = x2 at P is y - 1 = 2(x - 1)

∴ y = 2x - 1

Slope of tangent to y = x2 at P =`("dy"/"dx")_("at O" (1,1)) = 3xx1^2 = 3`

∴ equation of tangent to y = x3 at P is y - 1 = 3(x - 1)

∴ y = 3x - 2

We have to find angle between y = 2x - 1 and y = 3x - 2

Lines through origin parallel to these tagents are

y = 2x and y = 3x

∴ `"x"/1 = "y"/2 and "x"/1 = "y"/3`

These lines lie in XY-plane.

∴ the direction ratios of these lines are 1, 2, 0 and 1, 3, 0.

The angle θ between them is given by

cos θ = `((1)(1) + (2)(3) + (0)(0))/(sqrt(1^2 + 2^2 + 0^2)sqrt(1^2 + 3^2 + 0^2))`

`= (1 + 6 + 0)/(sqrt5 sqrt10)`

`= 7/sqrt50 = 7/(5sqrt2)`

∴ θ = `cos^-1(7/(5sqrt2))`

Hence, the required angles are 0 and `cos^-1(7/(5sqrt2))`

Concept: Vectors and Their Types
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×