Consider curve `y=f(x),` then if PT = Length of tangent PN= Length of normal TM=Length of subtangent MN=Length of sub normal, then (a)`T MdotM N=f^(prime)(x)` (b) `(P T)/(P N)=1/(f^(prime)(x))` (c)`C TdotP N=con s t a n t` (d) `P TdotT M=con s t a n t`

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

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.

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

Let P (h,K) be any point on curve y=f(x). Let tangent drawn at point P meets x-axis at T & normal at point P meets x-axis at N (as shown in figure) and m =(dy)/(dx)]_()(h,k)) = shope of tangent. (i) Length of tangent =PT =|K| sqrt(1+(1)/(m^(2))) (ii) Length of Normal =PN + |K| sqrt(1+m^(2)) (iii) Length subtangent = Projection of segment PT on x-axis =TM =|(k)/(m)| (iv) Length of subnormal =Projection of line segment PN on x-axis =MN =|Km| Find the product of length of tangent and length of normal for the curve y=x^(3)+3x^(2)+4x-1 at point x=0

The length of the normal at t on the curve x=a(t+sint), y=a(1-cos t), is

If f^(prime)(x)=sqrt(2x^2-1) and y=f(x^2),t h e n(dy)/(dx) at x=1 is (a)2 (b) 1 (c) -2 (d) none of these

For an A.P. show that t _(m) + t _(2n + m) = 2 t _(m +n)

If x=f(t )cost-f^(prime)(t)sint and y=f(t)sint+f^(prime)(t)cost ,t h e n((dx)/(dt))^2+((dy)/(dt))^2

Let P (h,K) be any point on curve y=f(x). Let tangent drawn at point P meets x-axis at T & normal at point P meets x-axis at N (as shown in figure) and m =(dy)/(dx)]_()(h,k)) = shope of tangent. (i) Length of tangent =PT =|K| sqrt(1+(1)/(m^(2))) (ii) Length of Normal =PN + |K| sqrt(1+m^(2)) (iii) Length subtangent = Projection of segment PT on x-axis =TM =|(k)/(m)| (iv) Length of subnormal =Projection of line segment PN on x-axis =MN =|Km| Determine 'p' such that the length of the subtangent and subnormal is equal for the curve y=e^(px) +px at the point (0,1)

If ("cos"(x-y))/("cos"(x+y))=m/n , then write the value of t a n x\ t a n ydot