Q. 315.0( 2 Votes )

Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point (x, y) is equal to the square of the difference of the abscissa and ordinate of the point.

Answer :

Abscissa means the x-coordinate and ordinate means the y-coordinate


Slope of tangent is given as square of the difference of the abscissa and ordinate


Difference of abscissa and ordinate is (x – y) and its square will be (x – y)2


Hence slope of tangent is (x – y)2


Slope of tangent of a curve y = f(x) is given by



Substitute x – y = z hence y = x – z



Differentiate x – z with respect to x






Using partial fraction for




Equating numerator


A(1 – z) + B(1 + z) = 1


Put z = 1


B = 1/2


Put z = -1


A = 1/2


Hence



Put in (a)



Integrate



x = 1/2(log(1 + z) + (–log(1 – z)) + c


2x = log(1 + z) – log(1 – z) + c


Using log a – log b = log



Resubstitute z = x – y



Now it is given that the curve is passing through origin that is (0, 0)


Hence (0, 0) will satisfy the curve equation (b)


Putting values x = 0 and y = 0 in (b)



c = 0


Put c = 0 back in equation (b)





e2x(1 – x + y) = (1 + x – y)


Hence equation of curve is e2x(1 – x + y) = (1 + x – y)


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Interactive Quiz on DIfferential CalculusFREE Class
Functional Equations - JEE with ease48 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses