Question:
Statement-1: The slope of the tangent at any point \(P\) on a parabola, whose axis is the axis of \(x\) and vertex is at the origin, is inversely proportional to the ordinate of the point \(P\).
Statement-2: The system of parabolas \(y^2=4ax\) satisfies a differential equation of degree 1 and order 1.
Statement-1: The slope of the tangent at any point \(P\) on a parabola, whose axis is the axis of \(x\) and vertex is at the origin, is inversely proportional to the ordinate of the point \(P\).
Statement-2: The system of parabolas \(y^2=4ax\) satisfies a differential equation of degree 1 and order 1.
Statement-2: The system of parabolas \(y^2=4ax\) satisfies a differential equation of degree 1 and order 1.
Show Hint
Differentiate y^2 = 4ax for Statement-1 and eliminate a for Statement-2; then ask whether the second explains the first.
Updated On: Oct 31, 2026
- Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1.
- Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
- Statement-1 is true; Statement-2 is false.
- Statement-1 is false; Statement-2 is true.
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The Correct Option is B
Solution and Explanation
Statement-1: for \(y^2=4ax\), \(2y\frac{dy}{dx}=4a\Rightarrow\frac{dy}{dx}=\frac{2a}{y}\). The slope is inversely proportional to the ordinate y, so Statement-1 is true.
Statement-2: eliminate the single parameter a using \(a=\frac{y\,y'}{2}\): \(y^2=4x\cdot\frac{y\,y'}{2}\Rightarrow y=2x\,y'\). This has order 1 and degree 1, so Statement-2 is true.
The DE \(y=2xy'\) describes the whole family (slope \(=\frac{y}{2x}\)); it does not by itself show that the slope varies as 1/y for a given parabola (that comes from differentiating a fixed parabola with constant a). So Statement-2 is not a correct explanation of Statement-1.
Answer: (2) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
Statement-2: eliminate the single parameter a using \(a=\frac{y\,y'}{2}\): \(y^2=4x\cdot\frac{y\,y'}{2}\Rightarrow y=2x\,y'\). This has order 1 and degree 1, so Statement-2 is true.
The DE \(y=2xy'\) describes the whole family (slope \(=\frac{y}{2x}\)); it does not by itself show that the slope varies as 1/y for a given parabola (that comes from differentiating a fixed parabola with constant a). So Statement-2 is not a correct explanation of Statement-1.
Answer: (2) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
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