Question

If x = 2 at, y = at², where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is 

 

Options

• 1/2a

• 1

• 2a

• none of these

Answer 1

(a) 1/2a

Here,

\[x = 2\text { at and y } = a t^2 \]

\[\text { Differentiating w . r . t . t, we get }\]

\[\frac{d x}{d t} = 2\text { a and } \frac{d y}{d t} = 2at\]

\[ \therefore \frac{d y}{d x} = \frac{2at}{2a} = t\]

\[\text { Differentiating w . r . t . x, we get }\]

\[\frac{d^2 y}{d x^2} = 1 \times \frac{dt}{dx} = \frac{1}{2a}\]

\[\text { Now,} \left[ \frac{d^2 y}{d x^2} \right]_{x = \frac{1}{2}} = \frac{1}{2a}\]