Question

If the subnormal at any point of curve y=3^1-k .X^k is of constant length,then 1/k equals to

Answer 1

The given curve is ,

y = $3^{1-k}\times x^{k}$

Differentiating both sides,we get

y'= $3^{1-k}$ × k ×$x^{k-1}$

Now, Length of subnormal is =y y'=p

→p=Length of subnormal=$3^{1-k}\times x^{k}\times3^{1-k}\times k\times x^{k-1}=k \times x^{2 k -1}\times 3^{2-2k}$ →as  ,$x^{a} \times x^{b} = x^{a+b}$

1/k = $\frac{x^{2 k -1}\times 3^{2-2k}}{\text {Length of subnormal}}$