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.

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.

At any point on the curve 2x^2y^2-x^4=c , the mean proportional between the abscissa and the difference between the abscissa and the sub-normal drawn to the curve at the same point is equal to or d in a t e (b) radius vector x-in t e r c e p toft a nge n t (d) sub-tangent

A curve is such that the mid-point of the portion of the tangent intercepted between the point where the tangent is drawn and the point where the tangent meets the y-axis lies on the line y=xdot If the curve passes through (1,0), then the curve is (a) ( b ) (c)2y=( d ) x^(( e )2( f ))( g )-x (h) (i) (b) ( j ) (k) y=( l ) x^(( m )2( n ))( o )-x (p) (q) (c) ( d ) (e) y=x-( f ) x^(( g )2( h ))( i ) (j) (k) (d) ( l ) (m) y=2(( n ) (o) x-( p ) x^(( q )2( r ))( s ) (t))( u ) (v)

Find the length of sub-tangent to the curve y=e^(x//a)

If x^m y^n=(x+y)^(m+n),prove (dy)/(dx)=y/xdot

A curve is defined parametrically be equations x=t^2a n dy=t^3 . A variable pair of perpendicular lines through the origin O meet the curve of Pa n dQ . If the locus of the point of intersection of the tangents at Pa n dQ is a y^2=b x-1, then the value of (a+b) is____

If the sub-normal at any point on y=a^(1-n)x^n is of constant length, then find the value of ndot