Problem 85 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A point on the parabola \(y^{2}=18 x\) at which the ordinate increases at twice the rate of the abscissa is (A) \((2,4)\) (B) \((2,-4)\) (C) \(\left(\frac{-9}{8}, \frac{9}{2}\right)\) (D) \(\left(\frac{9}{8}, \frac{9}{2}\right)\)

Step-by-Step Solution

Answer

Option (D): \(\left(\frac{9}{8}, \frac{9}{2}\right)\).

Step 1: Understand the Provided Equation

The given parabola is expressed by the equation \(y^2 = 18x\). This is a standard form of a parabola that opens to the right.

Step 2: Differentiate the Parabolic Equation

To find the rate of change, differentiate both sides of the equation \(y^2 = 18x\) with respect to \(t\): \[2y \frac{dy}{dt} = 18 \frac{dx}{dt}.\]

Step 3: Set the Rate Condition

The problem states that the ordinate (\(y\)) increases at twice the rate of the abscissa (\(x\)): \[\frac{dy}{dt} = 2 \frac{dx}{dt}.\]

Step 4: Substitute the Rate Condition

Substitute \(\frac{dy}{dt} = 2 \frac{dx}{dt}\) into the differentiated equation:\[2y(2 \frac{dx}{dt}) = 18 \frac{dx}{dt}.\]

Step 5: Simplify the Equation

Cancel \(\frac{dx}{dt}\) from both sides (assuming it is not zero), giving:\[4y = 18.\]

Step 6: Solve for y

Divide both sides by 4 to find \(y\): \[y = \frac{18}{4} = \frac{9}{2}.\]

Step 7: Use This y-value to Find x

Substitute \(y = \frac{9}{2}\) back into the original parabola equation \(y^2 = 18x\) to find x:\[(\frac{9}{2})^2 = 18x.\]

Step 8: Solve the Equation for x

Calculate and solve the equation:\[\frac{81}{4} = 18x\]\[x = \frac{81}{4 \times 18} = \frac{9}{8}.\]

Step 9: Determine the Correct Answer

The point that satisfies all conditions is \(\left(\frac{9}{8}, \frac{9}{2}\right)\), which corresponds to option (D).

Key Concepts

Parabola

The term "parabola" refers to a specific type of curve that appears in many mathematical contexts and real-world applications. It resembles a smooth, U-shaped curve that can open upwards, downwards, or sideways. The standard form of a parabola opening to the right is given by the equation

• For a parabola opening to the side, the equation is typically of the form:
  • \( y^2 = 4ax \)
• For a parabola opening upwards or downwards, it is in the form:
  • \( x^2 = 4ay \)

In the given problem, the equation \( y^2 = 18x \) portrays a parabola that opens to the right. This relates to many practical applications such as satellite dishes and vehicle headlights. Understanding the orientation and structure of a parabola helps in solving problems related to graphing and geometry.

Rate of Change

The concept of "Rate of Change" is pivotal in calculus and describes how a quantity changes over time. In the context of our exercise, this concept helps us examine how the positions along a parabola change.

These rates are defined in calculus as derivatives, and understanding them allows us to predict how one variable affects another.

• The rate of change is often expressed as \( \frac{dy}{dt} \) for the change in the vertical direction and \( \frac{dx}{dt} \) for the horizontal direction.
• In our exercise solution, the problem states that the ordinate (\( y \)) changes at twice the rate of the abscissa (\( x \)), expressed as \( \frac{dy}{dt} = 2 \frac{dx}{dt} \).

This condition closely ties the behavior of \( x \) and \( y \) as they move along the curve. Understanding these dynamic relationships is crucial not only in mathematics but in fields such as physics and economics.

Differentiation

Differentiation is a core concept in differential calculus, used primarily to determine the rate at which a variable changes. In the context of the given problem, it involves calculating how quantities change along the parabola.

The differentiation of the equation \(y^2 = 18x\) with respect to \(t\) is achieved by applying the chain rule:

• Differentiating the left side gives \(2y \frac{dy}{dt} \).
• The right side becomes \(18 \frac{dx}{dt} \).

By equating these, we establish a relationship allowing us to explore changes over time within the structure of the parabola. Solving these derived equations, as shown in the solution, helps in calculating specific points on the curve where given conditions are met. This understanding is foundational in advanced studies and real-world applications where systems change continuously.