Q96 / 180 · KCET 2015 · Mathematics
If \( x=c t \) and \( y=\frac{c}{t} \), find \( \frac{d y}{d t} \) at \( t=2 \)
- A\( \frac{1}{4} \)
- B\( 4 \)
- Correct answerC\( -\frac{1}{4} \)
- D\( 0 \)
Solution
\( x=c t \)
Given that, \( x c=t \) and \( y=\frac{c}{t} \)
Now \( \frac{d x}{d t}=c, \frac{d y}{d t}=\frac{-c}{t^{2}} \)
So, \( \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{-c}{c t^{2}}=\frac{-1}{t^{2}} \)
At \( t=2 \), we get
left. \( \left.\frac{d y}{d x}\right|_{t=2}=\frac{-1}{(2)^{2}}=\frac{-1}{4} \)
Now \( \frac{d x}{d t}=c, \frac{d y}{d t}=\frac{-c}{t^{2}} \)
So, \( \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{-c}{c t^{2}}=\frac{-1}{t^{2}} \)
At \( t=2 \), we get
left. \( \left.\frac{d y}{d x}\right|_{t=2}=\frac{-1}{(2)^{2}}=\frac{-1}{4} \)
Hence, the correct answer is option: 'C'.