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.

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

= e dy

= 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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