Problem 103 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: The locus of the centres of circles passing through the origin and cutting the circle \(x^{2}+y^{2}+6 x-\) \(4 y+2=0\) orthogonally is \(3 x-2 y+1=0\). Reason: The two circles \(x^{2}+y^{2}+2 g_{r} x+2 f_{1} y+c_{1}=0\) and \(x^{2}+y^{2}+2 g_{2} x+2 f_{2} y+c_{2}=0\) cut each other orthogonally if \(2 g_{1} g_{2}+2 f_{1} f_{2}=c_{1}+c_{2}\)

Step-by-Step Solution

Answer

The locus is the line \(3x - 2y + 1 = 0\).

Step 1: Understand the problem

We need to find the locus of centers of circles that pass through the origin and cut another given circle orthogonally. The circle's equation is \( x^2 + y^2 + 6x - 4y + 2 = 0 \).

Step 2: Analyze orthogonal circles condition

Two circles cut each other orthogonally if the sum of the products of their coefficients of \(x\) and \(y\) is equal to the sum of their constants. Mathematically: \(2g_1g_2 + 2f_1f_2 = c_1 + c_2\).

Step 3: Determine the parameters of the given circle

From the given circle \(x^2 + y^2 + 6x - 4y + 2 = 0\), identify the parameters: \(g_2 = 3\), \(f_2 = -2\), and \(c_2 = 2\).

Step 4: Define the general circle through the origin

Consider a circle that passes through the origin: \(x^2 + y^2 + 2g_1x + 2f_1y = 0\). Since the circle passes through the origin, the constant term \(c_1 = 0\).

Step 5: Apply orthogonal condition

Use the orthogonal circles condition: \(2g_1 \times 3 + 2f_1 \times (-2) = 0 + 2\). Simplifying this gives \(6g_1 - 4f_1 = 2\).

Step 6: Rearrange the equation to find the locus

Rearrange the equation \(6g_1 - 4f_1 = 2\) to express it as a linear equation. Dividing the whole equation by 2 gives \(3g_1 - 2f_1 = 1\). The locus is represented by the equation \(3x - 2y + 1 = 0\).

Key Concepts

Orthogonal Circles Condition

The concept of orthogonal circles is fascinating and vital in understanding their geometric relationships. Two circles are considered orthogonal if they intersect at right angles, meaning the tangents at their intersecting points are perpendicular. This property leads to the orthogonality condition used to determine when circles cut each other orthogonally.
To determine this condition mathematically, consider two circles defined by their equations:

• (First circle) \(x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0\)
• (Second circle) \(x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0\)

The orthogonal circles condition states:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
This equation arises when the sum of the products of corresponding coefficients of \(x\) and \(y\) is equal to the sum of the constants, \(c_1\) and \(c_2\).
Understanding and applying this condition allows us to determine whether given circles are orthogonal, which is crucial in calculations like finding the locus of centers of circles.

Equation of a Circle

The equation of a circle in the Cartesian plane is fundamental for describing its geometric properties. The standard form of a circle’s equation with a center at \((h, k)\) and radius \(r\) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
However, circles are often expressed in a general form, which is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Here, \(g, f,\) and \(c\) help define the circle's center and radius. The center \((h, k)\) of the circle can be identified as \((-g, -f)\) using the equation.
Moreover, the radius \(r\) is calculated as \(\sqrt{g^2 + f^2 - c}\) when the equation matches the general format. Understanding this equation is essential when solving problems related to a circle's location, size, and interaction with other geometric elements.
In exercises like the one provided, identifying the parameters from the circle's equation enables further analysis, such as applying orthogonal conditions or finding specific geometric loci.

Locus Determination

Determining the locus of a set of points or centers in geometry involves identifying the path or area containing all the positions fulfilling specific conditions. In our exercise, the locus refers to the centers of circles that intersect a given circle orthogonally and pass through a predefined point, like the origin.
To find the locus, follow these steps:

• Define the general circle passing through the required point, here it's the origin, leading to the equation: \(x^2 + y^2 + 2g_1x + 2f_1y = 0\).
• Apply the orthogonal condition using coefficients from the given circle and the general circle. This will yield an equation involving \(g_1\) and \(f_1\).
• Rearrange this equation to form a linear equation representing the locus of centers.

The result is a linear equation, like \(3x - 2y + 1 = 0\), which illustrates the geometric path of all possible centers fulfilling the problem's criteria.
This process of locus determination is essential to solving many geometry problems, allowing the visualization and identification of specific paths and configurations in the plane.