Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

The equation of locus of the point of intersection of tangents to the circle \(x^{2}+y^{2}=1\) at the points whose parametric angles differ by \(60^{\circ}\) is (A) \(3 x^{2}+3 y^{2}=1\) (B) \(x^{2}+y^{2}=3\) (C) \(3 x^{2}+3 y^{2}=4\) (D) none of these

Step-by-Step Solution

Answer

The locus of intersection is given by the equation \(3x^2 + 3y^2 = 4\), option (C).

Step 1: Understand the Problem

We need to find the locus of the intersection of tangents to a circle centered at the origin with radius 1. The circle's parametric points are defined by angles that differ by \(60^{\circ}\). Our task is to determine if the resulting locus satisfies one of the given equations.

Step 2: Write Parametric Equations

The parametric equations for the circle \(x^2 + y^2 = 1\) are \((x, y) = (\cos \theta, \sin \theta)\). If the angles differ by \(60^{\circ}\), then the points are \((\cos \theta, \sin \theta)\) and \((\cos(\theta + 60^\circ), \sin(\theta + 60^\circ))\).

Key Concepts

Circle Equations

A circle equation represents all the points that are equidistant from a central point, known as the center. In mathematics, when we talk about a circle on a plane, we focus on its equation in the standard form:

• \(x^2 + y^2 = r^2\)

Here,

• \((x, y)\) are the coordinates of any point on the circle,
• \(r\) is the radius of the circle,
• and the circle is centered at the origin \((0, 0)\).

The equation \(x^2 + y^2 = 1\) describes a circle with a radius of 1 centered at the origin. This serves as a simple foundation for exploring more complex geometric problems, such as finding points of tangents or using parametric equations to define points on the circle.

Tangent to a Circle

A tangent to a circle is a line that touches the circle at exactly one point. The uniqueness of a tangent is that it is perpendicular to the radius of the circle at the point of contact.

• The point where the tangent touches the circle is called the point of tangency.

In this problem, we consider the tangents that intersect at points defined by parametric angles differing by \(60^{\circ}\). The intersection of these tangents forms a locus of points, which is an important part of solving geometry problems.
Understanding tangents is crucial because it helps in finding the locus, or set of points, formed by such intersections, leading directly to a deeper understanding of the required geometric solution.

Parametric Equations

Parametric equations offer a way to describe the coordinates of points on curves, such as circles, by expressing the coordinates as functions of a parameter. In this case:

• For a circle with radius 1, the parametric equations are \(x = \cos \theta\) and \(y = \sin \theta\).

These equations provide an alternative representation that simplifies finding specific points on the circle, especially when dealing with angles like those differing by \(60^{\circ}\).

• The points in this problem are thus described as \((\cos \theta, \sin \theta)\) and \((\cos(\theta + 60^\circ), \sin(\theta + 60^\circ))\).

Parametric equations play a key role in solving complex geometry problems by allowing an easier and more visual interpretation of circular motion and related calculations.

Geometry Problem Solving

When tackling geometry problems, especially those involving circles, a structured approach is important. Understanding the essential concepts such as circle equations, parametric descriptions, and tangents is fundamental.

• Start by interpreting the problem's requirements, like defining the initial circle equation.
• Make use of parametric equations to deal with angles and circular motion.
• Investigate relationships like perpendicularity between tangents and radii.

Approach the solution step-by-step: first clarify the geometric shapes involved, and then work through the relationships and algebraic expressions they lead to.

• In this problem, confirming the differences in angles and using tangent properties helped derive the locus equation for the intersection of tangents.

Geometry problem solving is about translating visual interpretations into mathematical expressions, which can then be evaluated for solutions that meet the problem's criteria.