Question:
If $x = a \sin t - b \cos t$ and $y = a \cos t + b \sin t$, and it is given that $\frac{d^2y}{dx^2} = 0$, then:
If $x = a \sin t - b \cos t$ and $y = a \cos t + b \sin t$, and it is given that $\frac{d^2y}{dx^2} = 0$, then:
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Whenever the second derivative is zero, the curve is a straight line.
Updated On: May 16, 2026
- $y = \text{constant}$
- $y = x$
- $y = ax + b$
- $y = x + c$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
The second derivative $\frac{d^2y}{dx^2} = 0$ indicates that the rate of change of the slope is zero, meaning the relation between $x$ and $y$ is linear.
Step 2: Meaning
Integrating the condition $\frac{d^2y}{dx^2} = 0$ twice with respect to $x$ provides the functional form of the relationship.
Step 3: Analysis
First integration: $\frac{dy}{dx} = m$ (where $m$ is a constant slope). Second integration: $y = mx + c$ (where $c$ is a constant).
Step 4: Conclusion
Among the given options, $y = ax + b$ is the only expression that represents the standard general equation of a straight line. Final Answer: (C)
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