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

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

(A) \( \theta \) is the angle of \( OP \) from the positive direction of the \( x \)-axis.

Step 1: Understand the problem context

We have an ellipse defined by the equation \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) and a point \( P(a \cos \theta, b \sin \theta) \) on this ellipse. We need to determine what the angle \( \theta \) represents in this context.

Step 2: Recall the parametric form of an ellipse

The parametric equations of an ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) are \( x = a \cos \theta \) and \( y = b \sin \theta \), where \( \theta \) is the parameter that determines the position of a point \((x,y)\) on the ellipse.

Step 3: Determine the physical meaning of \( \theta \)

In the parametric form \( x = a \cos \theta \) and \( y = b \sin \theta \), the angle \( \theta \) is the angle from the positive \( x \)-axis to the line \( OP \), where \( O \) is the origin and \( P(a \cos \theta, b \sin \theta) \) is the point on the ellipse. Thus, it defines the line segment from the center of the ellipse to the point \( P \).

Step 4: Evaluate the options

Option (A) states that \( \theta \) is the angle of \( OP \) from the positive \( x \)-axis, which matches the interpretation derived from the parametric equations. Other options either don't apply or change the context by considering a different point \( Q \), which is not relevant here since \( b \sin \theta \) is not replaced by \( a \sin \theta \).

Step 5: Conclude the correct choice

From our analysis, Option (A) directly corresponds to the derived meaning of \( \theta \) as the angle of the line \( OP \) from the positive direction of the \( x \)-axis.

Key Concepts

Parametric Equation

A parametric equation provides a way to express the coordinates of points on a curve using equations that depend on an additional parameter, often represented by \( \theta \) or \( t \). For an ellipse described by \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), the parametric form is uniquely beneficial.
* The **x-coordinate** is defined as \( x = a \cos \theta \).
* The **y-coordinate** is \( y = b \sin \theta \).
Here, \( a \) and \( b \) are the semi-major and semi-minor axes, respectively, and \( \theta \) is the angle. These parametric equations give us a simple way to pinpoint any point on the ellipse by just varying \( \theta \). As \( \theta \) varies from 0 to \( 2\pi \), the point \( (x, y) \) traces the complete perimeter of the ellipse.
Since parametric equations translate an algebraic equation into a geometrical path followed by a point, they are especially useful in coordinate geometry to understand the locus of points on curves like ellipses.

Angle on Ellipse

In the context of an ellipse, the angle \( \theta \) has a specific geometric interpretation. It describes the angle in standard position from the positive direction of the x-axis to the line that connects the origin \( (0,0) \) to a point \( P(a \cos \theta, b \sin \theta) \) on the ellipse.
* **Standard Position**: When angles are measured from the positive x-axis as a baseline.
* **Line OP**: Represents the line segment from the origin to the point \( P \) on the ellipse, where \( P \) is defined parametrically by \( (a \cos \theta, b \sin \theta) \).
This means that for any point \( P \) on the ellipse, \( \theta \) corresponds directly to this central angle, showing how far the line OP has rotated starting from the positive x-axis.
So, in terms of the ellipse, \( \theta \) serves as a key rotational marker for identifying phase angles along the ellipse's path.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebra to investigate geometrical problems through the coordinate system. The equation of an ellipse, \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), exemplifies how algebraic expressions can represent geometric shapes:

• **Centre and Axes**: The centre of the ellipse lies at the origin (0,0). The x-axis and y-axis serve as geometric constructs to define the ellipse's orientation.
• **Axes lengths**: Here, \( a \) and \( b \) denote the semi-major and semi-minor axes. They represent the longest and shortest distances from the centre to the perimeter of the ellipse.
• **Shape Description**: Describes an ellipse as a locus of points where the sum of distances from any point on the ellipse to two fixed points (foci) remains constant.

In coordinate geometry, the solutions to algebraic equations like this can yield profound insights not just into location but also length, distance, and even transformation. On an ellipse, these relationships help in understanding various local and global properties, integrating calculus principles when deriving properties like arc length or surface area.