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Sagot :
To find [tex]\(\frac{dy}{dx}\)[/tex] given the parametric equations [tex]\(x = a t^2\)[/tex] and [tex]\(y = 2at\)[/tex], we'll follow a systematic approach.
1. Express [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in terms of [tex]\(t\)[/tex]:
[tex]\[ x = a t^2 \][/tex]
[tex]\[ y = 2a t \][/tex]
2. Calculate the derivatives [tex]\(\frac{dx}{dt}\)[/tex] and [tex]\(\frac{dy}{dt}\)[/tex]:
- Differentiating [tex]\(x\)[/tex] with respect to [tex]\(t\)[/tex]:
[tex]\[ \frac{dx}{dt} = \frac{d}{dt}(a t^2) = 2at \][/tex]
- Differentiating [tex]\(y\)[/tex] with respect to [tex]\(t\)[/tex]:
[tex]\[ \frac{dy}{dt} = \frac{d}{dt}(2at) = 2a \][/tex]
3. Find [tex]\(\frac{dy}{dx}\)[/tex] using the chain rule:
[tex]\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \][/tex]
4. Substitute the derivatives [tex]\(\frac{dy}{dt}\)[/tex] and [tex]\(\frac{dx}{dt}\)[/tex] into the equation:
[tex]\[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{2a}{2at} = \frac{1}{t} \][/tex]
Therefore, the answer is:
[tex]\[ \frac{dy}{dx} = \frac{1}{t} \][/tex]
So, the correct choice is (a) [tex]\(\frac{1}{t}\)[/tex].
1. Express [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in terms of [tex]\(t\)[/tex]:
[tex]\[ x = a t^2 \][/tex]
[tex]\[ y = 2a t \][/tex]
2. Calculate the derivatives [tex]\(\frac{dx}{dt}\)[/tex] and [tex]\(\frac{dy}{dt}\)[/tex]:
- Differentiating [tex]\(x\)[/tex] with respect to [tex]\(t\)[/tex]:
[tex]\[ \frac{dx}{dt} = \frac{d}{dt}(a t^2) = 2at \][/tex]
- Differentiating [tex]\(y\)[/tex] with respect to [tex]\(t\)[/tex]:
[tex]\[ \frac{dy}{dt} = \frac{d}{dt}(2at) = 2a \][/tex]
3. Find [tex]\(\frac{dy}{dx}\)[/tex] using the chain rule:
[tex]\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \][/tex]
4. Substitute the derivatives [tex]\(\frac{dy}{dt}\)[/tex] and [tex]\(\frac{dx}{dt}\)[/tex] into the equation:
[tex]\[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{2a}{2at} = \frac{1}{t} \][/tex]
Therefore, the answer is:
[tex]\[ \frac{dy}{dx} = \frac{1}{t} \][/tex]
So, the correct choice is (a) [tex]\(\frac{1}{t}\)[/tex].
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