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

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.

Find the curve for which the length of normal is equal to the radius vector.

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

Find the lengths of tangent, subtangent, normal and subnormal to y^2=4ax at (at^2,2at)

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.

The curve for which the length of the normal is equal to the length of the radius vector is/are (a) circles (b) rectangular hyperbola (c) ellipses (d) straight lines

What will be the equivalent focal length for two lenses of focal length f_1 and f_2 in contact ?

If the length of sub-normal is equal to the length of sub-tangent at any point (3,4) on the curve y=f(x) and the tangent at (3,4) to y=f(x) meets the coordinate axes at Aa n dB , then the maximum area of the triangle O A B , where O is origin, is 45/2 (b) 49/2 (c) 25/2 (d) 81/2

Find the curve for which the perpendicular from the foot of the ordinate to the tangent is of constant length.

The parametric equation of a curve is given by, x=a(cos t+log tan(t/2)) , y=a sin t. Prove that the portion of its tangent between the point of contact and the x-axis is of constant length.