Problem 4
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 \(\mathrm{P}\) Statement-2: The system of parabolas \(y^{2}=4 a x\) satisfies a differential equation of degree 1 and order 1 . (a) Statement- 1 is true; Statement- 2 is true; Statement- 2 is a correct explanation for statement- 1 . (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for statement-1. (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is false; Statement- 2 is true.
Step-by-Step Solution
Verified Answer
(b) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
1Step 1: Understanding the Tangent Slope
For the parabola of the form \(y^2 = 4ax\), the general formula for the slope of the tangent at any point \((x_1, y_1)\) is given by \(\frac{dy}{dx} = \frac{y_1}{2a}\). From Statement-1, the slope is inversely proportional to \(y_1\), which suggests a relationship \( \frac{dy}{dx} \propto \frac{1}{y_1} \). Simplifying using the equation of the tangent, we see that \( \frac{y_1}{2a} \equiv \frac{1}{y_1} \), matching the inversion. Thus, Statement-1 is true.
2Step 2: Describing the Parabola System
For the parabolas defined by \(y^2 = 4ax\), differentiate both sides to obtain \(2y\frac{dy}{dx} = 4a\). By rearranging, the differential equation becomes \(y\frac{dy}{dx} = 2a\), which is a first-order, first-degree differential equation. This confirms Statement-2 as true.
3Step 3: Analyzing Explanation Relationship
Statement-2 describes a characteristic of the parabola's differential equation but does not directly explain the proportionality aspect in Statement-1. This suggests that while both statements are true, Statement-2 does not serve as a direct explanation for Statement-1's claim about slope proportionality.
Key Concepts
ParabolaDifferential EquationTangent SlopeJEE Mathematics Problems
Parabola
A parabola is one of the simple and classic forms of a conic section, easily recognizable by its U-shape. In mathematics, it's often described using equations that show the relationship between its points. A common form for parabolas is the equation \( y^2 = 4ax \), which describes a parabola that opens to the right, with its vertex at the origin \((0, 0)\).
If the equation changes, the parabola rotates or shifts accordingly. In real-world terms, parabolas are what you see in paths of projectiles like thrown balls or water in fountains.
Understanding parabolas is key to solving various mathematics problems, especially in calculus where we analyze changes along its curve.
If the equation changes, the parabola rotates or shifts accordingly. In real-world terms, parabolas are what you see in paths of projectiles like thrown balls or water in fountains.
Understanding parabolas is key to solving various mathematics problems, especially in calculus where we analyze changes along its curve.
Differential Equation
A differential equation relates a function with its derivatives. In simple terms, it tells us how a function changes at any given point. For the parabola \(y^2 = 4ax\), we derive a differential equation by differentiating both sides: \(2y \frac{dy}{dx} = 4a\).
By rearranging, we get \(y \frac{dy}{dx} = 2a\), which is a first-order, first-degree differential equation.
By rearranging, we get \(y \frac{dy}{dx} = 2a\), which is a first-order, first-degree differential equation.
- First-Order: Involves the first derivative of the function.
- First-Degree: The highest power of the derivative is one.
Tangent Slope
When we talk about the slope of a tangent line to a parabola, we're describing how steep the line is when it just touches the curve. For a given parabola \(y^2 = 4ax\), the slope of the tangent at any point \((x_1, y_1)\) is \( \frac{dy}{dx} = \frac{y_1}{2a} \).
In Statement-1, the claim that the slope is inversely proportional to the ordinate \(y_1\) implies a relationship of \( \frac{dy}{dx} \propto \frac{1}{y_1} \).
This concept of inverse proportionality is essential because it tells us how the steepness of the tangent changes with respect to its location on the curve. Understanding tangent slopes helps in visualizing and calculating angles that a curve forms with the horizontal axis at specific points.
In Statement-1, the claim that the slope is inversely proportional to the ordinate \(y_1\) implies a relationship of \( \frac{dy}{dx} \propto \frac{1}{y_1} \).
This concept of inverse proportionality is essential because it tells us how the steepness of the tangent changes with respect to its location on the curve. Understanding tangent slopes helps in visualizing and calculating angles that a curve forms with the horizontal axis at specific points.
JEE Mathematics Problems
The Joint Entrance Examination (JEE) is a highly competitive test for aspiring engineers in India. JEE Main Mathematics problems often include advanced topics such as conic sections, calculus, and differential equations.
Mastering these topics requires a clear understanding of fundamental concepts, like those of parabolas and tangent slopes, and their related algebraic and calculus principles.
To excel in JEE, students must practice solving problems that explore these areas:
Mastering these topics requires a clear understanding of fundamental concepts, like those of parabolas and tangent slopes, and their related algebraic and calculus principles.
To excel in JEE, students must practice solving problems that explore these areas:
- Understanding and Manipulating Equations: Being able to derive and interact with equations helps in solving complex problems.
- Recognizing Patterns and Relationships: Identifying relationships such as inversely proportional components is key.
- Applying Calculus Techniques: Calculus questions, especially involving derivatives and differential equations, are prevalent.
Other exercises in this chapter
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The differential equation which represents the family of curves \(y=c_{1} e^{c_{2} x}\), where \(c_{1}\), and \(c_{2}\) are arbitrary constants, is (a) \(y^{\pr
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