In the curve `x^(m+n)=a^(m-n)y^(2n)` , prove that the `m t h` power of the sub-tangent varies as the `n t h` power of the sub-normal.

Show that for the curve b y^2=(x+a)^3, the square of the sub-tangent varies as the sub-normal.

In the curve x^(m)y^(n)=k^(m+n)(m,n, k gt0) prove that the portion of the tangent intercepted between the coordinate axes is divided at its point of contact in a constant ratio,

For the curve y=a1n(x^2-a^2) , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

Show that the length of the tangent to the curve x^(m)y^(n)=a^(m+n) at any point of it, intercepted between the coordinate axes is divided internally by the point of contact in the ratio m:n.

Determine p such that the length of the sub-tangent and sub-normal is equal for the curve y=e^(p x)+p x at the point (0,1)dot

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is (a) ( b ) (c) y=( d ) e^(( e ) (f) K(( g ) (h) x-1( i ))( j ))( k ) (l) (m) (b) ( n ) (o) y=( p ) e^(( q ) (r) K e (s))( t ) (u) (v) (c) ( d ) (e) y=( f ) e^(( g ) (h) K(( i ) (j) x-2( k ))( l ))( m ) (n) (o) (d) None of these

If (x+y)^(m+n)=x^(m)y^(n) , then the value of (dy)/(dx) is -

If x^m y^n=(x+y)^(m+n),p rov et h a t(dy)/(dx)=y/xdot

If a^(m^n)= (a^m)^n , then express m in terms of n.

For the curve y=f(x) prove that (length of normal)^2/(length of tangent)^2 =length of sub-normal/length of sub-tangent