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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
Solution
(a) 1/2a
Here,
\[x = 2\text { at and y } = a t^2 \]
\[\text { Differentiating w . r . t . t, we get }\]
\[\frac{d x}{d t} = 2\text { a and } \frac{d y}{d t} = 2at\]
\[ \therefore \frac{d y}{d x} = \frac{2at}{2a} = t\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = 1 \times \frac{dt}{dx} = \frac{1}{2a}\]
\[\text { Now,} \left[ \frac{d^2 y}{d x^2} \right]_{x = \frac{1}{2}} = \frac{1}{2a}\]
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