Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the chord of contact of tangents from a point on the circle \(x^{2}+y^{2}=a^{2}\) to the circle \(x^{2}+y^{2}=b^{2}\) touches the circle \(x^{2}+y^{2}=c^{2}\), then \(a, b, c\) are in (A) A. P. (B) G. P. (C) H. P. (D) none of these

Step-by-Step Solution

Answer

\(a, b, c\) are in G.P.

Step 1: Review the Problem Setup

We have three circles defined by their equations. The outer circle has the equation \(x^{2} + y^{2} = a^{2}\), the middle circle has \(x^{2} + y^{2} = b^{2}\), and the inner circle has \(x^{2} + y^{2} = c^{2}\). The chord of contact of the tangents from a point on the outer circle to the middle circle touches the inner circle.

Step 2: Equation of the Chord of Contact

The chord of contact of the tangents from a point \((x_1, y_1)\) on the circle \(x^{2} + y^{2} = a^{2}\) to the circle \(x^{2} + y^{2} = b^{2}\) is given by the equation \(x_1x + y_1y = b^{2}\).

Step 3: Condition for Tangency

For the chord \(x_1x + y_1y = b^{2}\) to touch the circle \(x^{2} + y^{2} = c^{2}\), the distance from the center (0, 0) to the line must equal the radius of the inner circle. The distance from the center to the line is \(\frac{|b^{2}|}{\sqrt{x_1^{2} + y_1^{2}}} = \frac{b^{2}}{a}\), and it must equal \(c\).

Step 4: Formulate the Relationship

Setting the distance equal to the radius of the inner circle, we have \(\frac{b^{2}}{a} = c\). Simplifying this gives \(b^{2} = ac\). This equation suggests that \(a, b^{2}, c\) are in a relationship where \(b^{2}\) is a geometric mean, implying they are in G. P.

Key Concepts

Chord of Contact

The concept of the chord of contact relates to two tangent lines drawn from an external point to a circle. When you draw these tangents, they touch the circle at two distinct points. The line joining these points of tangency is known as the chord of contact.
In this exercise, we consider tangents drawn from a point located on one circle to a second circle. The equation of this chord of contact is derived using the formula: \[ x_1x + y_1y = b^2 \] where

• \((x_1, y_1)\) is the external point on the first circle \(x^2 + y^2 = a^2\)
• \(b^2\) is the circle to which the tangents are drawn

Understanding the chord of contact helps to determine the relationships between the distances and properties of the involved circles. It's a pivotal concept in exploring how geometric properties align, especially when dealing with multiple concentric circles.

Tangents to Circle

A tangent to a circle is a line that touches the circle at exactly one point. This singular point of contact ensures that the tangent never crosses into the circle itself, maintaining its uniqueness.
Tangents play a critical role in the solution to our exercise. Here, the tangents originate from a point on a circle with radius \(a\) and extend to a separate circle with radius \(b\). The chord of contact is then established between these points of tangency.
To satisfy the condition of tangency with a third circle, the distance from the center of the circle to the chord of contact must be equal to the radius of this third circle. In mathematical terms, when this condition is satisfied:\[ \frac{b^2}{a} = c \]It implies that the tangency condition is fulfilled, which is instrumental in determining geometrical alignments or relationships among the circles.

Geometric Progression

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio."
In this exercise, analyzing the relationship between the radii of the circles forms a key part of understanding their geometric layout. We are given that if a chord of contact, defined by the tangents from an outer circle to a middle circle, touches a third inner circle, then these circles' radii must satisfy a particular condition: \( b^2 = ac \).
This equation denotes that the radius squared of the middle circle \(b^2\) is the geometric mean of the other two radii \(a\) and \(c\). Consequently, this forms a geometric progression where \(a, b, c\) can be seen in sequence such that:

• \(a\) is the first term
• \(b^2\) is effectively the ratio squared between terms
• \(c\) is the final term

Understanding this relationship is crucial as it interconnects algebraic properties with tangible geometric concepts.