Problem 69 - A Complete Resource Book in Mathematics for JEE Main

Question

A circle whose centre coincides with the origin having radius ' \(a\) ' cuts \(x\)-axis at \(A\) and \(B\). If \(P\) and \(Q\) are two points on the circle whose parametric angles differ by \(2 \theta\), then the locus of the intersection point of \(A P\) and \(B Q\) is (A) \(x^{2}+y^{2}+2 a y \tan \theta=a^{2}\) (B) \(x^{2}+y^{2}-2 a y \tan \theta=a^{2}\) (C) \(x^{2}+y^{2}+2 a y \cot \theta=a^{2}\) (D) none of these

Step-by-Step Solution

Answer

The locus is given by \(x^2 + y^2 + 2ay \tan \theta = a^2\). Option (A) is correct.

Step 1: Understanding the problem

We need to find the locus of the intersection point of lines formed by points on a circle with center at origin and radius 'a' at different parametric angles. Lines are formed using the circle's intersection points A and B on the x-axis and two points P and Q whose parametric angles differ by \(2\theta\).

Step 2: Define the points on the circle

The equation of the circle is \(x^2 + y^2 = a^2\). The intersection points with the x-axis are \(A = (a, 0)\) and \(B = (-a, 0)\). If \(P\) has a parametric angle \(\phi\), then \(P = (a \cos \phi, a \sin \phi)\), and \(Q\) has \(\phi + 2\theta\), thus \(Q = (a \cos(\phi + 2\theta), a \sin(\phi + 2\theta))\).

Step 3: Parametric equations for lines AP and BQ

Line \(AP\) can be described with parametric form: \((1 - t)a + ta \cos \phi, ta \sin \phi\), and line \(BQ\) with \((1 - t)(-a) + ta \cos(\phi + 2\theta), ta \sin(\phi + 2\theta)\), where \(t\) is the parameter.

Step 4: Finding intersection of lines AP and BQ

Set the two parametric lines equal to find their intersection points. This requires \(a(1 - t) + ta \cos \phi = -(a - ta \cos(\phi + 2\theta))\) and \(ta \sin \phi = ta \sin(\phi + 2\theta)\).

Step 5: Solve for locus equation

From the equality conditions derived, solve the simultaneous equations to eliminate parameters, obtaining a condition in \(x, y\). Substituting and simplifying reveals \(x^2 + y^2 + 2ay \tan \theta = a^2\). This matches option (A).

Key Concepts

Parametric Equations

When exploring problems in geometry, specifically in context like the intersection of lines and circular geometry, parametric equations become very useful. They help us describe points in a coordinate system where the coordinates themselves are set based on a parameter, rather than being defined in terms of each other directly.

For a point on a circle, parametric equations can be used effectively. A circle with a center at the origin and radius \(a\) can be expressed with:

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

Here, \(\theta\) is a parameter that represents the angle the radius makes with the positive x-axis, measured in radians. By changing \(\theta\), you can describe every point on the circle.

For example, in the problem, you need to define points \(P\) and \(Q\) using parametric equations. If \(P\) is defined by an angle \(\phi\), then \(P\) would be \((a \cos \phi, a \sin \phi)\). If \(Q\)'s angle is \(\phi + 2\theta\), then \(Q = (a \cos(\phi + 2\theta), a \sin(\phi + 2\theta))\). This representation allows us to handle operations related to movements and transformations easily in terms of a parameter, simplifying solutions of geometry problems.

Intersection of Lines

In the context of geometric problems involving lines and intersections, understanding how to find the intersection of lines is crucial. The intersection point is essentially the set of coordinates that satisfy the equations of both lines simultaneously. Using parametric forms is one efficient way to express lines when they are part of a dynamic geometric setup.

For instance, say you're given line segments \(AP\) and \(BQ\), where \(A\) and \(B\) are fixed points (in this case, these are the points where the circle cuts the x-axis), and \(P\) and \(Q\) are points that vary along the circle. The parametric equations for these lines can be expressed as:

• \(AP: (1 - t)a + ta \cos \phi, ta \sin \phi\)
• \(BQ: (1-t)(-a) + ta \cos(\phi + 2\theta), ta \sin(\phi + 2\theta)\)

By equating these, you are setting their x and y expressions equal to solve for the intersection. Unlike standard linear equations, here every point on a line is defined relative to parameter \(t\), which runs from 0 to 1.

Solving these simultaneous equations results in conditions that eliminate parameters, helping us derive the locus equation for the intersection point. This process helps in determining the path on which such intersections exist, often forming a new geometric shape or curve.

Circular Geometry

Circular geometry brings a wealth of interesting properties and theorems that tie together angles, circles, and different geometric figures. In the present context, the circle helps explore loci—paths traced by moving points—and their intersections.

Understanding the basics of circular geometry involves knowing that:

• A circle's standard equation is \(x^2 + y^2 = r^2\) for a circle centered at the origin with radius \(r\).
• Intersecting a circle with an axis (like the x-axis) leads to straightforward intersection points based on radius values.

In our exercise, these intersections are part of fixed points \(A\) and \(B\), which aid in developing lines with other circle points like \(P\) and \(Q\).

When solving such problems, it often reduces to translating polar (angle-radius form) into Cartesian (x-y form), via parametric equations. This comes to life when dealing with tangents, secants, or any line interacting with circular arcs. Circular geometry intertwines with concepts like parametric equations when you measure angles or define positions on the circumference.

The problem of finding intersection loci combines these principles to explore new territories in geometry, showing just how interconnected these fields are in broader mathematical contexts.