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

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

The answer is (B) G. P.

Step 1: Define the Chord of Contact

The chord of contact of tangents from a point \(x_1, y_1\) to the circle \(x^2 + y^2 = b^2\) is given by the equation \(xx_1 + yy_1 = b^2\).

Step 2: Identify the Point on the Circle

Assume the point \(x_1, y_1\) lies on the circle \(x^2 + y^2 = a^2\). Thus, we have the equation \(x_1^2 + y_1^2 = a^2\).

Step 3: Condition for Tangency

The chord of contact \(xx_1 + yy_1 = b^2\) touches the circle \(x^2 + y^2 = c^2\) if the perpendicular distance from the center of \(x^2 + y^2 = c^2\) to the line is equal to the radius of the circle. This gives us the condition: \(|b^2|/\sqrt{x_1^2 + y_1^2} = c\).

Step 4: Simplify the Condition

Using the equation \(x_1^2 + y_1^2 = a^2\) from Step 2, substitute into the tangency condition to get \(|b^2|/a = c\).

Step 5: Determine the Relationship

The condition \(|b^2|/a = c\) can be rewritten as \(b^2 = ac\). This implies that the terms \(a, b, c\) must satisfy \(b^2 = ac\), which is the definition of numbers being in geometric progression (G.P.).

Step 6: Final Conclusion: Check Options

Based on the derived relationship, \(a, b, c\) are in G.P. Thus, the correct answer is (B) G. P.

Key Concepts

Chord of Contact

When studying circle geometry, the chord of contact plays an important role. It is the line segment that joins the points of tangency of two tangents drawn from an external point to a circle.
For simplification, if the external point is \(x_1, y_1\) and the equation of the circle is \(x^2 + y^2 = r^2\), the equation for the chord of contact can be denoted as \(xx_1 + yy_1 = r^2\). This equation describes the line containing all the points that form chords tangent to the given circle from \(x_1, y_1\).
This concept is crucial in solving various geometric problems involving tangents because it provides a bridge between an external point and a circle, showing how tangents can be geometrically related to the circle’s parameters.

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.
For example, the sequence \(a, ar, ar^2, ar^3,...\) illustrates a G.P. with \(r\) as the common ratio.
In the context of circle geometry, when we say three values \(a, b, c\) are in geometric progression, it implies the relationship \(b^2 = ac\). Essentially, the square of the middle term equals the product of the other two terms.

• This relationship is not just abstract. In problems involving circles, recognizing a G.P. can be key to unlocking solutions, especially when dealing with properties like tangency and symmetry.

Tangency Condition

In circle geometry, the tangency condition is essential for understanding how lines interact with circles. This condition states that a line is tangent to a circle if it touches the circle at exactly one point.
For a line to be tangent, the distance from the center of the circle to the line must be equal to the radius of the circle. Mathematically, for a circle \(x^2 + y^2 = c^2\) and a line \(ax + by + c = 0\), the condition for tangency can be assessed by: \[ \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} = r \] where \(r\) is the radius.
This equation ensures that the perpendicular distance between the circle center and line equals the radius, signifying a perfect touch – or tangency.

• This concept is critical when solving problems that involve determining if lines are tangents and employing this condition can lead to uncovering deeper relationships between circles and lines.