Q. 165.0( 1 Vote )

Find the curve for which the intercept cut – off by a tangent on the x – axis is equal to four times the ordinate of the point of contact.

Answer :

Let P(x,y) be the point of contact of tangent and curve y = f(x).

It cuts the axes at A and B so, the equation of the tangent at P(x,y)


Y – y = (X – x)


Putting X = 0


Y – y = (0 – x)


Y = y – x


So, A(0, y – x)


Now, putting Y = 0


0 – y = (X – x)


X = x – y


So, B(x – y,0)


Given, intercept on x – axis = 4× ordinate


x – y = 4y


y + 4y = x


+ 4 =


= – 4


We can see that it is a linear differential equation.


Comparing it with


P = , Q = – 4


I.F = e∫Pdy


= edy


= e – logy


=


Solution of the given equation is given by


x × I.F = ∫Q × I.F dy + logc


x × () = ∫ – 4 × dy + logc


= – 4 log y + log c


= log y – 4 + logc


= log c y – 4


= c y – 4


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