Problem 21 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the locus of a point which moves so that the line joining the points of contact of the tangents drawn from it to the circle \(x^{2}+y^{2}=b^{2}\) touches the circle \(x^{2}+\) \(y^{2}=a^{2}\), is 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

The numbers \(a, b, c\) are in arithmetic progression (A. P.).

Step 1: Understanding the Problem

We need to find the relationship between \(a\), \(b\), and \(c\) given the setup of a geometric locus problem. The problem describes a scenario involving two circles and tangents, with the condition that the locus of a point leads to another circle. The circle \(x^2 + y^2 = b^2\) is the original, and \(x^2 + y^2 = a^2\) is touched by the locus circle \(x^2 + y^2 = c^2\).

Step 2: Using Tangents to a Circle

The length of the tangent from a point \((h, k)\) to a circle with center \((0, 0)\) and radius \(b\) is \(\sqrt{h^2 + k^2 - b^2}\). Since the locus forms a circle that touches the circle \(x^2 + y^2 = a^2\), the radius of the locus circle combined with its center’s distance gives us a condition.

Step 3: Setting Up the Locus Condition

The locus we are examining results in the circle \(x^2 + y^2 = c^2\), which means that the line connecting the tangential points forms the diameter of this locus circle. Thus, the radius of this locus circle (\(c\)) will be related to both \(a\) and \(b\).

Step 4: Applying the Circle Touch Condition

If the circle of the locus \(x^2 + y^2 = c^2\) touches the circle \(x^2+y^2=a^2\) externally, the sum of the radii \(a\) and \(c\) equals the distance from the origin to the center of the locus, plus the radius of the original circle \(b\). Thus, \(a = b + c\).

Step 5: Analyzing the Relationship

From \(a = b + c\), observe that this is an arithmetic progression, where \(a - b = c\). Hence, the numbers \(a, b, c\) are in arithmetic progression.

Key Concepts

Circles

Understanding circles is key in geometry, especially when we discuss problems involving the locus of a point. In essence, a circle is all the points in a plane that are a fixed distance from a given point, known as the center. This fixed distance is called the radius. Let's explore some points about circles:

• The equation of a circle with center at the origin \(0, 0\) and radius \(r\) is given by \(x^2 + y^2 = r^2\).
• A circle centered at \(h, k\) has the equation \((x - h)^2 + (y - k)^2 = r^2\).

In our problem, we have two circles, \(x^2 + y^2 = b^2\) and \(x^2 + y^2 = a^2\). Here, both circles are centered at the origin but have different radii \(b\) and \(a\) respectively.In geometric problems, understanding how lines or points interact with a circle is crucial, especially when considering the locus of a point, which often forms another circle. This forms the foundation of many geometric exercises.

Tangents

A tangent to a circle is a straight line that touches the circle at exactly one point. When we discuss tangents in relation to circles, we focus on their unique properties:

• The tangent at any point of a circle is perpendicular to the radius at the point of contact.
• If you draw two tangents to a circle from an external point, these tangents are equal in length.

In our problem, the point from which tangents are drawn to the circle \(x^2 + y^2 = b^2\) assists in establishing the locus of a point. Understanding tangents helps us connect the points of contact that form the diameter of a new circle, which is our locus circle \(x^2 + y^2 = c^2\).The properties of tangents also allow us to deduce relationships and conditions about distances and radii in geometrical setups, emphasizing their significance in these exercises. Thus, correctly applying the tangent properties is essential in reaching the correct conclusion about the relationship among \(a, b, c\).

Geometric Progression

Geometric Progression (G.P) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this problem, we need to determine how \(a, b,\) and \(c\) relate. However, the solution shows that they are not in geometric progression; instead, they are in Arithmetic Progression (A.P).Understanding the difference between geometric and arithmetic progression is critical:

• In a G.P, each term is the previous one multiplied by a constant. For example, 2, 4, 8 is a G.P with a common ratio of 2.
• In an A.P, each term is obtained by adding a constant difference to the previous term, like in our problem where \(a = b + c\).
• The relationship found, \(a - b = c\), indicates an arithmetic pattern where the difference between consecutive elements is constant.

The exercise determined that \(a, b, c\) form an arithmetic progression, offering insights into the general understanding and application of progression sequences in geometry.