Problem 109 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The equation of a tangent to the parabola, \(x^{2}=8 y\), which makes an angle \(\theta\) with the positive direction of \(x\)-axis, is: (a) \(y=x \tan \theta+2 \cot \theta\) (b) \(y=x \tan \theta-2 \cot \theta\) (c) \(x=y \cot \theta+2 \tan \theta\) (d) \(x=y \cot \theta-2 \tan \theta\)

Step-by-Step Solution

Answer

Option (a): \(y = x \tan \theta + 2 \cot \theta\).

Step 1: Identify the Parabola Equation

The given parabola equation is \( x^2 = 8y \). This is a standard form of a parabola that opens upwards.

Step 2: Find the Slope of the Tangent

The slope of the tangent to the parabola that makes an angle \( \theta \) with the positive x-axis will be \( \tan \theta \).

Step 3: Standard Tangent Equation to Parabola

The equation of a tangent to the parabola \( x^2 = 4ay \) is \( y = mx + \frac{a}{m} \), where \( m \) is the slope. For \( x^2 = 8y \), \( a = 2 \), so the tangent equation becomes \( y = mx + \frac{2}{m} \).

Step 4: Substitute the Slope Value

Since the slope \( m \) is \( \tan \theta \), substitute \( m = \tan \theta \) into the tangent equation: \( y = x \tan \theta + \frac{2}{\tan \theta} \).

Step 5: Simplify the Equation

Simplify the equation \( y = x \tan \theta + \frac{2}{\tan \theta} \) to \( y = x \tan \theta + 2 \cot \theta \).

Step 6: Identify the Correct Option

The simplified equation matches option (a): \( y = x \tan \theta + 2 \cot \theta \).

Key Concepts

Tangent Equation to a Parabola

To understand the tangent equation to a parabola, we first need to look at the standard form of the parabola. For the given equation \( x^2 = 8y \), it is a form that represents a parabola opening upwards. In general, for any parabola \( x^2 = 4ay \), the geometry dictates that the standard tangent line equation is \( y = mx + \frac{a}{m} \), where \( m \) symbolizes the slope of the tangent.
When we substitute the value \( a = 2 \) into the equation for our specific parabola, we arrive at the equation \( y = mx + \frac{2}{m} \). This formula encapsulates the relationship between the slope and the upward opening parabola.

• Standard parabola equation: \( x^2 = 4ay \)
• Tangent line equation for parabola: \( y = mx + \frac{a}{m} \)
• For \( x^2 = 8y \), \( a = 2 \), derived tangent equation: \( y = mx + \frac{2}{m} \)

Slope of Tangent

The slope of a tangent line to a curve at a given point provides information on how steep the line is as it touches the curve at that particular point. For parabolas such as \( x^2 = 8y \), if a tangent line forms an angle \( \theta \) with the positive x-axis, the slope of the tangent can be expressed as \( \tan \theta \).
This is because the slope \( m \) of the tangent line is the tangent of the angle it makes with the x-axis. Hence, in the equation \( y = mx + \frac{2}{m} \), replacing \( m \) with \( \tan \theta \) gives us a more specific equation for a tangent line to the parabola:

• Relation between angle, \( \theta \), and slope: \( m = \tan \theta \)
• Tangent line equation with slope \( \tan \theta \): \( y = x \tan \theta + \frac{2}{\tan \theta} \)

Angle with X-axis

The angle that a tangent line makes with the positive x-axis is crucial for determining the orientation and slope of the tangent. This angle, denoted as \( \theta \), directly influences the slope formula \( \tan \theta \), which describes how the tangent line rises or falls.
When the tangent equation is set as \( y = x \tan \theta + 2 \cot \theta \), it reflects how the line interacts with the parabola using \( \theta \). The term \( \tan \theta \) showcases the direction and steepness of the tangent, while \( 2 \cot \theta \) adjusts vertically based on the specific point of tangency. Understanding this allows for the prediction of the line's orientation:

• Angle \( \theta \) defines the line's slope: \( \tan \theta \)
• Small \( \theta \) results in gentle slopes, while larger \( \theta \) leads to steeper lines
• Correct tangent equation identifies the interplay between slope and angle: \( y = x \tan \theta + 2 \cot \theta \)