Question

if the tangent at the point p(X1,y1) on the curve x^m×y^n=a^m+n meets the axis at Aand B .the ratio in which P divides AB is

​

Answer 1

Ans=

x

m

y

n

=a

m+n

.

Take log both sides

mlogx+nlogy=(m+n)loga

Differentiating w.r.t x, we get

m⋅

x

1

+n⋅

y

1

dx

dy

=0

∴

dx

dy

=−

n

m

x

y

Hence the equation of the tangent is

Y−y=−

n

m

x

y

(X−x)

⇒myX+nxY=(m+n)⋅xy...(1)

Let the tangent (1) meet the axes in points A and B.

Putting Y=0, we get A=[

m

m+n

x,0]

Putting X=0 we get B=(0,

m

m+n

y)

Point of contact P is (x,y).

Let P divide AB in the ratio λ:1

x=

λ+1

λ.0+1⋅

n

m+n

x

⇒(λ+1)x=(

n

m

+1)x

⇒λ+1=

n

m

+1

∴λ=

n

m