Problem 107 - A Complete Resource Book in Mathematics for JEE Main

Question

Assertion: The equation of the circle passing through the point \((2 a, 0)\) and whose radical axis is \(x=\frac{a}{2}\) with respect to the circle \(x^{2}+y^{2}=a^{2}\), will be \(x^{2}+y^{2}-2 a x=0\). Reason: The equation of radical axis of two circles \(x^{2}\) \(+y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0\) and \(x^{2}+y^{2}+2 g_{2} x+2 f y+c_{2}\) \(=0\) is \(2\left(g_{1}-g_{2}\right) x+2\left(f_{1}-f_{2}\right) y+\left(c_{1}-c_{2}\right)=0\).

Step-by-Step Solution

Answer

The assertion is correct; the circle's equation is \(x^2 + y^2 - 2ax = 0\).

Step 1: Understand the given problem

We need to find the equation of a circle that passes through the point \((2a, 0)\), has a radical axis \(x=\frac{a}{2}\) with respect to the circle \(x^2+y^2=a^2\), and show that this circle can be expressed as \(x^2 + y^2 - 2ax = 0\).

Step 2: Analyze the given circle

The circle \(x^2+y^2=a^2\) has its center at the origin \((0, 0)\) and radius \(a\).

Step 3: Identify the properties of the circle passing through (2a, 0)

Assume a general form for the circle as \(x^2+y^2 + 2gx + 2fy + c = 0\). Given it passes through point \((2a, 0)\), substitute in the equation: \((2a)^2 + (0)^2 + 2g(2a) + 2f(0) + c = 0\), simplifying to \(4a^2 + 4ag + c = 0\).

Step 4: Use the radical axis condition

The radical axis with circle \(x^2+y^2=a^2\) is given by \(x=\frac{a}{2}\). The radical axis of two circles \(x^2+y^2+2gx+2fy+c=0\) and \(x^2+y^2=a^2\) simplifies to \(2gx+2fy+(c-a^2)=0\). By substituting \(x = \frac{a}{2}\), it becomes \(g + \frac{1}{2}(c - a^2) = 0\).

Step 5: Solve for constants using the radical axis

From the radical axis condition \(g + \frac{1}{2}(c - a^2) = 0\), solve for \(c\): \(g + \frac{c}{2} - \frac{a^2}{2} = 0\), resulting in \(c = a^2 - 2g\). Using the point condition \(4a^2 + 4ag + c = 0\), substitute \(c = a^2 - 2g\) into it, you get an expression for \(g\).

Step 6: Equation of the circle derived

Substitute the values back into the general equation \(x^2 + y^2 + 2gx + 2fy + c = 0\), we find \(g=-a\) and \(c=0\). The derived equation for the circle is \(x^2 + y^2 - 2ax = 0\).

Step 7: Verification of assertion

The equation \(x^2 + y^2 - 2ax = 0\) is shown to pass through \((2a, 0)\) and matches the conditions of the radical axis, thus verifying the assertion.

Key Concepts

Radical Axis

The radical axis is a powerful concept in circle geometry. It deals with the properties of two circles and their interaction with each other. If you have two circles with equations

• \(x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0\) and
• \(x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0\),

the line that connects all the points of equal power with respect to these two circles is called the radical axis.

The equation for the radical axis can be derived as:\[2(g_1 - g_2)x + 2(f_1 - f_2)y + (c_1 - c_2) = 0\]In simpler terms, the radical axis is the locus of points where the power of both circles is the same. We can think of it as a line where the tangents from a point on this line to each circle are the same length.

For our exercise, the radical axis is given by the vertical line \(x = \frac{a}{2}\), which tells us that on this line, the power relative to both circles is balanced.

General Equation of a Circle

When we talk about the general equation of a circle in coordinate geometry, we refer to the equation that can represent any circle on a plane. The standard form of this equation is:

• \(x^2 + y^2 + 2gx + 2fy + c = 0\)

In this equation:

• \(g\) and \(f\) represent the coordinates of the circle's center in modified form, specifically, the center is \((-g, -f)\).
• The radius \(r\) can be calculated by the formula: \(r = \sqrt{g^2 + f^2 - c}\).

In our problem, we are given that the circle passes through the point \((2a, 0)\), and has a specific radical axis. By substituting the conditions provided (including the radical axis) into this general form, we can systematically solve for \(g\), \(f\), and \(c\), to find that the equation simplifies to \(x^2 + y^2 - 2ax = 0\). This shows that

• \(g = -a\)
• \(c = 0\), and
• \(f = 0\) as there is no \(y\) term.

This indicates the circle's center is at \((a, 0)\) with a radius of \(a\).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. Here, positions are determined on a plane using numbers, making it an essential tool for dealing with shapes like circles, lines, and other conic sections through algebraic equations.
For example, the positioning of a circle on the plane is easily described by its center \((h, k)\) and its radius \(r\). The relation between a circle's equation and its graphical representation is crucial, as we used in this exercise.

By analyzing given conditions such as the point a circle passes through or its radical axis, we can derive specific properties of circles and lines and solve for unknown values.

In this context, coordinate geometry allows us to seamlessly blend algebraic manipulation with geometric intuition, simplifying the process of visualizing and solving complex problems related to circles. This combination is particularly noticeable when we substitute coordinates into equations to deduce the nature of geometric figures or when we equate geometrical axioms to algebraic expressions.