Question:
The general solution of $4xp^{2} = (3x - a)^{2}$, $p = \frac{dy}{dx}$, is}
The general solution of $4xp^{2} = (3x - a)^{2}$, $p = \frac{dy}{dx}$, is}
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When $p^2$ is isolated, take the root to turn a higher-degree equation into a simple integration problem.
Updated On: Jul 3, 2026
- $(y + c)^{2} = x(x - a)^{2}$
- $(x - a)^{3} = y^{2} + c$
- $(y + c)^{2} = (x - a)^{2}$
- $(y - c)^{2} = (x + a)^{2}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
This equation is solvable for $p$ by taking the square root of both sides.
Step 2: Meaning
$2\sqrt{x} p = \pm(3x - a) \Rightarrow p = \frac{dy}{dx} = \pm \frac{3x - a}{2\sqrt{x}}$.
Step 3: Analysis
Separate variables and integrate: $dy = \pm \frac{3x - a}{2\sqrt{x}} dx$. This becomes $dy = \pm (\frac{3}{2}x^{1/2} - \frac{a}{2}x^{-1/2}) dx$. Integrating gives $y + c = \pm (x^{3/2} - ax^{1/2})$.
Step 4: Conclusion
Factoring out $\sqrt{x}$ gives $y + c = \pm \sqrt{x}(x - a)$. Squaring both sides to remove the sign yields $(y + c)^{2} = x(x - a)^{2}$. Final Answer: (A)
This equation is solvable for $p$ by taking the square root of both sides.
Step 2: Meaning
$2\sqrt{x} p = \pm(3x - a) \Rightarrow p = \frac{dy}{dx} = \pm \frac{3x - a}{2\sqrt{x}}$.
Step 3: Analysis
Separate variables and integrate: $dy = \pm \frac{3x - a}{2\sqrt{x}} dx$. This becomes $dy = \pm (\frac{3}{2}x^{1/2} - \frac{a}{2}x^{-1/2}) dx$. Integrating gives $y + c = \pm (x^{3/2} - ax^{1/2})$.
Step 4: Conclusion
Factoring out $\sqrt{x}$ gives $y + c = \pm \sqrt{x}(x - a)$. Squaring both sides to remove the sign yields $(y + c)^{2} = x(x - a)^{2}$. Final Answer: (A)
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