Problem 100 - A Complete Resource Book in Mathematics for JEE Main

Question

The differential equation of all non-vertical lines in a plane is [2002] (A) \(\frac{d^{2} y}{d x^{2}}=0\) (B) \(\frac{d^{2} x}{d y^{2}}=0\) (C) \(\frac{d y}{d x}=0\) (D) \(\frac{d x}{d y}=0\)

Step-by-Step Solution

Answer

The correct answer is (A) \(\frac{d^{2}y}{dx^{2}} = 0\).

Step 1: Understand the problem

We need to determine which option correctly represents the differential equation for all non-vertical lines in a plane. Non-vertical lines have a constant slope.

Step 2: Recall the slope of a line

In analytical geometry, for straight lines, the slope of the line is constant. Thus, the derivative \( \frac{dy}{dx} \), which represents the slope, should also be constant.

Step 3: Translate into a differential equation

Since the slope \( \frac{dy}{dx} \) is constant for any non-vertical line, its derivative with respect to \( x \), \( \frac{d^{2}y}{dx^{2}} \), must be equal to zero.

Step 4: Identify the correct option

The differential equation that ensures the slope \( \frac{dy}{dx} \) remains constant is \( \frac{d^{2}y}{dx^{2}} = 0 \), which matches option (A).

Key Concepts

Slope of a Line

The slope of a line, often denoted as the letter "m," is a fundamental concept in analytical geometry. It's a measure of the steepness or the angle of inclination of a line on a Cartesian plane. The slope is calculated as the ratio of the "rise" (the change in the y-coordinate) to the "run" (the change in the x-coordinate) between any two points on a line.
This is mathematically expressed as:

• Slope (m) = \( \frac{y_2 - y_1}{x_2 - x_1} \)

Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two distinct points on the line.
In the context of differential equations and analytical geometry, the slope corresponds to the first derivative \( \frac{dy}{dx} \) when dealing with lines defined by their equations in the form \( y = mx + c \), where "c" is the y-intercept. For non-vertical lines, this slope remains constant across the entire span of the line.

Non-Vertical Lines

Non-vertical lines are lines that are neither horizontal nor vertical; they have a definite slope, which may be positive or negative. Unlike vertical lines, which have undefined slopes due to a division by zero in their slope formula, non-vertical lines have a slope that is a finite number.
In differential equations, non-vertical lines can be described by equations of the form \( y = mx + c \), where "m" represents the constant slope and "c" the y-intercept. The slope, given by \( \frac{dy}{dx} \), remains constant as long as the line is non-vertical.

• They have a constant first derivative.
• Their characteristic differential equation \( \frac{d^{2}y}{dx^{2}} = 0 \) ensures this constancy of slope.

It's essential to understand that only non-vertical lines can be described by this kind of differential equation, reinforcing their unique property of constant slope.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, uses a coordinate system to explore geometrical shapes and their properties. This branch of mathematics is instrumental for visualizing algebraic equations in a geometric context, leading to a profound comprehension of shapes, sizes, and relative positions within a coordinate plane.
It links geometric principles and algebra by using algebraic equations to represent geometrical figures, allowing the study of figures across multiple dimensions:

• For lines, the slope-indicating equations such as \( y = mx + c \) are fundamental.
• Differential equations, like \( \frac{d^{2}y}{dx^{2}} = 0 \), connect the calculus concepts to provide deeper insights into algebraic curves and their behavior.

Through analytical geometry, problems spanning from basic line equations to complex curve integrations can be addressed, making it a versatile and valuable tool in both theoretical and applied mathematics.