Problem 12 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(P(a \cos \theta, b \sin \theta)\) is a point on an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), then ' \(\theta\) ' is (A) angle of \(O P\) line from positive direction of \(x\)-axis ( \(O\) is origin) (B) angle of \(O Q\) line from positive direction of \(x\)-axis \([\) when \(Q\) is \((a \cos \theta, a \sin \theta)]\) (C) it depends on the point \(P\) (D) none of the above

Step-by-Step Solution

Answer

Option A is correct.

Step 1: Understand the Point on Ellipse

The point \( P(a \cos \theta, b \sin \theta) \) given in the problem is provided to be on an ellipse defined by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This ensures that the coordinates of \( P \) satisfy the equation, confirming they adhere to the properties of an ellipse.

Step 2: Verify that P is on the Ellipse

Plug the coordinates of point \( P(a \cos \theta, b \sin \theta) \) into the ellipse equation:\[ \frac{(a \cos \theta)^2}{a^2} + \frac{(b \sin \theta)^2}{b^2} = 1. \] Simplifying both terms we get: \( \cos^2 \theta + \sin^2 \theta = 1 \), which is a true identity.

Step 3: Interpret the Meaning of \( \theta \)

The expression \( P(a \cos \theta, b \sin \theta) \) describes a parametric form for points on the ellipse. The angle \( \theta \) in this context is used to express the location of a point \( P \) as it varies along the ellipse's path.

Step 4: Analyze the Angle of the Line OP

The line \( OP \), where \( O \) is the origin and \( P(a \cos \theta, b \sin \theta) \), forms an angle \( \theta \) with the positive direction of the \( x \)-axis, as this is how the parametric representation of a point on an ellipse is defined.

Key Concepts

Understanding Parametric Equations in the Context of an Ellipse

When working with ellipses, parametric equations are a powerful tool. They differ from regular equations because they use parameters to express coordinates. For an ellipse, the parametric equations are usually given as:

• \( x = a \cos \theta \)
• \( y = b \sin \theta \)

These equations describe the position of any point on the ellipse using the parameter \( \theta \), which often represents an angle. The variables \(a\) and \(b\) represent the semi-major and semi-minor axes of the ellipse, respectively. Using parametric equations can make it simpler to work out various coordinates because they allow you to describe the curve in terms of a single variable. By adjusting \(\theta\), you can trace out a point around the ellipse, describing the entire shape flexibly. This parameterization helps in understanding and graphing ellipses, especially when dealing with angles and line properties.

Interpreting the Angle of a Line in Parametric Forms

When exploring the properties of lines in geometry, the angle they make with the axes can be significant. Specifically, consider the line \(OP\), where \(O\) is the origin and \(P\) is a point \((a \cos \theta, b \sin \theta)\). The parameter \(\theta\) directly indicates the angle formed between this line and the positive direction of the x-axis. This line's angle is a feature of the parametric representation. It shows the relationship between the ellipse's geometry and the point \(P\)'s position. In this case, \(\theta\) acts as a simple way to control how points move around the ellipse relative to the x-axis. This idea can be crucial for graphing and understanding elliptical shapes within coordinate systems.

Exploring Trigonometric Identity in the Context of Ellipses

In trigonometry, an identity is an equation that is true for all values of the involved variables. In the context of ellipses, trigonometric identities ensure relationships are preserved as points move around the curve. When verifying a point \( (a \cos \theta, b \sin \theta) \) on an ellipse, the identity \( \cos^2 \theta + \sin^2 \theta = 1 \) comes into play. This identity is pivotal because it confirms that the parametric equations used indeed form an ellipse when substituted back into its standard equation. For the given ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), plugging in the parametric expressions and simplifying shows how the trigonometric identity ensures that coordinates adhere to the required elliptical shape. Thus, trigonometric identities provide confidence in working with these mathematical forms, confirming properties of points that lie on an ellipse.