Problem 97 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If, \(A, B, C\) be the centres of three coaxal circles and \(t_{1}, t_{2}, t_{3}\) be the tangents to them from any point, then \(B C \cdot t_{1}^{2}+C A \cdot t_{2}^{2}+A B \cdot t_{3}^{2}\) (A) 0 (B) 1 (C) \(-1\) (D) none of these

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Understanding the Problem

We need to evaluate the expression \(B C \cdot t_{1}^{2}+C A \cdot t_{2}^{2}+A B \cdot t_{3}^{2}\), where \(t_1, t_2, t_3\) are tangents from a point to three coaxial circles with centers \(A, B, C\). We need to find which value among the given options the expression evaluates to.

Step 2: Using the Power of a Point Theorem

The power of a point theorem states that the power of a point relative to a circle entails the relation \(t^2 = OP^2 - r^2\), where \(OP\) is the distance from the point to the circle center and \(r\) is the radius. Applying to all circles, we have: \(t_1^2 = PA^2 - R_1^2\), \(t_2^2 = PB^2 - R_2^2\), \(t_3^2 = PC^2 - R_3^2\).

Step 3: Relating Tangents to Distances

For coaxial circles, the pivotal property is that the differences between their squared radii remain constant across any pair of circles. Hence, if \(A, B, C\) form a line, \(BC = AB + AC\) or any other following properties influence each tangent squared term resulting in telescopic cancellations when summed.

Step 4: Substituting Relations

Since \(t_1^2, t_2^2, t_3^2\) can be exactly expressed with differences relative to coaxial properties, substituting these relations into \(B C \cdot t_{1}^{2}+C A \cdot t_{2}^{2}+A B \cdot t_{3}^{2}\) effectively simplifies, resulting in the equation eliminating radii terms, resulting in perfect zeroes through subtraction.

Step 5: Evaluating the Expression

After substituting, the expression \(B C \cdot t_{1}^{2}+C A \cdot t_{2}^{2}+A B \cdot t_{3}^{2}\) simplifies to zero due to the coaxial alignment and cancellation of all respective squared radius relations.

Key Concepts

Coaxial Circles

Coaxial circles, or coaxal circles, refer to a system of circles that share the same line, known as their axis. The axis can either be a real line or an imaginary line if the circles are considered in complex planes. A unique characteristic of coaxial circles is that every pair of circles in the system has the same radical axis—a line that is perpendicular to the line joining their centers and is equidistant from all pairs of intersecting circles.
Understanding coaxial circles is important since their setup creates predictable geometric properties, such as constant differences between their squared radii regardless of the specific pair of circles in the system. This property simplifies the calculation of expressions involving tangents from a common external point to these circles. Such simplifications play a vital role in solving problems related to the Power of a Point and telescopic sum cancellations as seen in the exercise.

Circle Geometry

Circle geometry focuses on properties and relationships related to circles, involving points, lines, and angles related to circles. A core concept here is the Power of a Point theorem, a powerful result used for solving problems with tangents and distances.
This theorem states that for any point outside a circle, the square of the length of a tangent drawn from that point to the circle is equal to the difference between the square of the distance from the point to the center and the square of the radius. If a point is outside multiple circles, as with coaxial circles, this theorem applies repeatedly, simplifying expressions involving tangents significantly.

• To apply the Power of a Point in problems, determine the distance from the point to each circle's center and know the radii.
• Another critical factor in circle geometry is understanding how to leverage properties of symmetry and aligned centers (as seen in the exercise) for problem simplification.

This knowledge equips you with the tools needed to tackle complex problems involving coaxial circle systems.

Telescopic Sum

A telescopic sum is a series of numbers, terms, or expressions where most intermediate terms cancel each other out when expanded in a sequence. This concept is invaluable for simplifying complicated arithmetic or algebraic expressions, often reducing them to simpler terms or even zero.
In the context of the given exercise, the expression involving tangent lengths squared simplifies due to the property of coaxial circles. Each tangent squared term is paired with distances and radius squares that align in such a way that they form a telescopic sum. Here’s how it typically works:

• The matching algebraic terms cancel due to properties relating tangents and distances in coaxial circles.
• For example, in the expression: \(BC \cdot t_1^2 + CA \cdot t_2^2 + AB \cdot t_3^2\), the squared tangent expressions derive from distance equations \(PA^2, PB^2,\) etc., with coaxial properties ensuring cancellation.

This results in an expression that reduces entirely, often to zero, signifying a precise relationship dependent profoundly on coaxial axes geometry.