7.2.5 Angle between two curves

Definition . Angle between two curves, if they intersect, is defined as the acute angle between the tangent lines to those two curves at the point of intersection. For the given curves, at the point of intersection using the slopes of the tangents, we can measure the acute angle between the two curves. Suppose y m x and y m x are two lines, then the acute angle θ between these lines is given by, tan θ = m m m ...

( ) where m and m are finite. Remark (i) If the two curves are parallel at x y ) , then m (ii) If the two curves are perpendicular at x y ) and if m and m exists and finite then m m = − . Example . Find the angle between y and y .

Let us now find the point of intersection of the two given curves. Equating x ) we get, x = . Therefore, the point of intersection is   . Let θ be the angle between the curves.

The slopes of the curves are as follows : For the curve y = x , dx = x . Let m = at   = . For the curve y = ( x − , dx = x − Let m = at   = − . Fig.

. – – θ y = ( x – ) y = x Using ( ), we get tan θ = −− Hence, θ = tan −  . Example . Find the angle between the curves y and x at their points of intersection ( , ) and ( , ).

Let us now find the slopes of the curves. Let m be the slope of the curve y , then m = dy = . Let m be the slope of the curve x , then m = dy = . Let θ and θ be the angles at ( , ) and ( , ) respectively.

At ( , ) , we come across the indeterminate form of ×∞ in the denominator of tan θ and so we follow the limiting process. tan θ = lim x y lim x y = ¥ which gives θ = tan ( ) − ∞= π . At ( , ) , m tan θ = = which gives θ = tan −  . Fig.