Q. 255.0( 1 Vote )

Find the equation of the curve passing through the point (1, 1), if the perpendicular distance of the normal at P(x, y) to the curve from the origin is equal to the distance of P from the x - axis.

Answer :

Here, P(x, y) is the point on the curve y = f(x)

We can find the equation of normal at P by



Here, The equation of normal at P is





- - - (1)


It is given that the distance from the origin is equal to the distance from the x - axis


Therefore,






…(i)


Let, F(x, y) =


Now, put x = λx and y = λy in F(x, y)



Taking λ as common from both numerator and denominator



We know, If the degree of the λ in function F(x, y) then it is said Homogenous function


So, the given differential equation is Homogenous differential equation.


Put y = vx in equation(i)


Differentiation y w.r.t x , we get


…(iii)


Now, Compare the equation (i) and (iii), we get



Taking x2 as common from both numerator and denominator







Integrating both sides, we get



Let v2 + 1 = t


On differentiating we get,


2v dv = dt, So



log t = - log x + log C


Putting the value of t , we get


Log|v2 + 1| + log|x| = log|c|


log|x(v2 + 1)| = log|C|


Now, putting back the value of v = y/x ,




- - - (iv)


And, The curve is passing through (1, 1)


Then, x = 1 and y = 1


(1)2 + 1 = C(1)


C = 2


Put the value of C in equation (iv)


y2 + 1 = x(2)


y2 = 2x - 1


Hence, The curve is y2 = 2x - 1


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 CalculusInteractive Quiz on DIfferential CalculusInteractive Quiz on DIfferential Calculus50 mins
Interactive Quiz on Differential Calculus | Check YourselfInteractive Quiz on Differential Calculus | Check YourselfInteractive Quiz on Differential Calculus | Check Yourself56 mins
Functional Equations - JEE with easeFunctional Equations - JEE with easeFunctional 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
caricature
view all courses