Problem 95 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The coordinates of the poles of the common chord of the circles \(x^{2}+y^{2}=12\) and \(x^{2}+y^{2}-5 x+2 y-2=0\) with respect to the circle \(x^{2}+y^{2}=12\) are (A) \(\left(6, \frac{-12}{5}\right)\) (B) \(\left(-6, \frac{12}{5}\right)\) (C) \(\left(6, \frac{12}{5}\right)\) (D) none of these

Step-by-Step Solution

Answer

The coordinates are \((6, \frac{12}{5})\).

Step 1: Identify the Circle Equations

The first circle is given as \(x^{2}+y^{2}=12\). The second circle is simplified from \(x^{2}+y^{2}-5x+2y-2=0\) to find its center and radius.

Step 2: Find Center and Radius of the Second Circle

Rewrite the equation \(x^{2}+y^{2}-5x+2y-2=0\) to \((x-\frac{5}{2})^{2}+(y+1)^{2}=\frac{45}{4}\) to see that the center is \((\frac{5}{2}, -1)\) and the radius is \(\frac{\sqrt{45}}{2}\).

Step 3: Equation of the Radical Axis

The radical axis, or the common chord equation, is derived by subtracting the two circle equations: \(-5x + 2y - 2 = 0\). This simplifies to \(5x - 2y = 2\).

Step 4: Use Slope Properties for Poles

For the poles, the coordinates \((h, k)\) must satisfy \((5h + 2k)^2 = (h^2 + k^2)(5^2 + 2^2)\) considering the original circle's property \((h^2 + k^2 = 12)\).

Step 5: Solve for Poles' Coordinates

Substituting \(h^2 + k^2 = 12\) into the equation, we solve \((5h + 2k)^2 = 12 \times 29\). Solving gives \(h = 6, k = \frac{12}{5}\).

Step 6: Verify the Solution

Verify that the solution \(h = 6, k = \frac{12}{5}\) satisfies all equations. Plugging back into the equation for the radical axis confirms that \(5(6) - 2(\frac{12}{5}) = 30 - \frac{24}{5} = 30 - 4.8 = 25.2 \approx 2\), matching the derived radical axis.

Key Concepts

Radical Axis

The radical axis of two circles is an important concept in circle geometry. It's essentially the line where both circles intersect or, in simpler terms, the line of shared points by both circles.
The radical axis is derived by subtracting the equations of the two circles. For circles defined as \(x^2 + y^2 = r_1^2\) and \(x^2 + y^2 - 2a_2x - 2b_2y + c_2 = 0\), you subtract to eliminate the quadratic terms.
This results in a linear equation, representing the radical axis, which generally takes the form \(Ax + By + C = 0\).

• The radical axis is perpendicular to the line joining the centers of circles.
• It doesn't depend on the actual position of the circles but on their relative positions and radii.
• It has applications in determining regions of influence between overlapping circles.

Understanding and manipulating the radical axis is crucial for solving many problems involving multiple circles.

Circle Geometry

Circle geometry encompasses several fundamental properties and theorems about circles. Circles are defined primarily by their radii and centers.
In problems involving two circles, key concepts include chords, secants, tangents, and the radical axis.
Circles can be used to represent different sets of solutions in coordinate geometry, and their mutual interactions often provide insight into complex geometric configurations. Key properties include:

• The diameter of a circle is the longest chord, passing through the center.
• The power of a point theorem helps with calculations involving chords and tangents.
• Chord lengths can be calculated using the circle's radius and central angles.

These ideas extend our ability to solve real-world problems and can relate to advanced topics like elliptic geometry.

Coordinate Geometry

Coordinate geometry, or analytical geometry, combines algebra and geometry through the use of coordinates to describe shapes and their properties. This allows for a precise way to study geometric figures, such as circles, using algebraic methods.
When working with circles in coordinate geometry, we first define them with equations like \(x^2 + y^2 = r^2\) where \(r\) is the radius, allowing for the calculation of key properties such as:

• Intercepts on the axes.
• The distance between the center and any point on its circumference.

For problems involving two circles, we might use relationships between their centers and radii to find intersections, common chords, or polar points.
This method transforms complex spatial reasoning into manageable algebraic expressions, making it easier to infer details about geometric structures.