Problem 47 - A Complete Resource Book in Mathematics for JEE Main

Question

The common chord of the circle \(x^{2}+y^{2}+8 x+4 y-5=\) 0 and a circle passing through the origin and touching the line \(y=x\), passes through the fixed point (A) \(\left(\frac{5}{12}, \frac{5}{12}\right)\) (B) \(\left(\frac{5}{12}, \frac{-5}{12}\right)\) (C) \(\left(\frac{-5}{12}, \frac{5}{12}\right)\) (D) none of these

Step-by-Step Solution

Answer

The solution leads to knowing that the point (A) with coordinates \((\frac{5}{12}, \frac{5}{12})\) is valid and the common chord passes through this position.

Step 1: Identify the given circle equations

The given equation of a circle is \[ x^2 + y^2 + 8x + 4y - 5 = 0 \] We need to identify its properties. We can rewrite this equation in the standard form of a circle: \[ (x + 4)^2 + (y + 2)^2 = 25 \]. It represents a circle with center (-4, -2) and radius \(\sqrt{25}=5\).

Step 2: Determine the second circle

The second circle passes through the origin (0,0) and touches the line y=x. Any line tangent to a circle implies a perpendicular radius at the point of tangency. Maintaining this tangent condition ensures the circle's center is equidistant from the x=y line.

Step 3: Equation of second circle via parametric formation

The general equation for a circle passing through (0,0) is \(x^2 + y^2 + 2gx + 2fy = 0\), where \(f, g\) determine the center. Since it touches the line y = x, substitute y = x in terms to find that relation, ensuring it's consistent within the equation.

Step 4: Equation of the common chord

A common chord can be established by subtracting equations of intersecting circles of form \(S - S_1 = 0\)yielding the required common chord function.

Step 5: Finding fixed point using chord

For differing circle selections, note that subtracting parameters guarantees a fixed passage through a fixed coordinate. Find intersection of derived paths and identify options presented.

Step 6: Verification of options

Check by calculating simplified representations and including verifications against given pairs (A), (B), (C), or (D), focusing particularly on simultaneous equation outcomes to ensure matching or negation.

Key Concepts

Circle Equation

In geometry, a circle is often represented by an equation in the coordinate plane. The most common form is the standard form:

• \((x - h)^2 + (y - k)^2 = r^2\)

where

• \((h, k)\) are the coordinates of the circle's center, and
• \(r\) is the radius of the circle.

When we have terms like \(x^2 + y^2 + Dx + Ey + F = 0\), we can convert it to the standard form by completing the square. In this case, transforming \(x^2 + y^2 + 8x + 4y - 5 = 0\) yields the standard form \((x + 4)^2 + (y + 2)^2 = 25\), indicating a circle centered at \((-4, -2)\) with a radius of 5.
This transformation involves finding perfect squares for both \(x\) and \(y\) terms by adjusting constants. Understanding how to convert equations helps identify circle attributes easily.

Common Chord

A common chord between two circles is a line where the circles intersect. It plays a crucial role in understanding circle geometry. You can find it using the method of subtracting the equations of the two circles. Given two circles, \(S = x^2 + y^2 + 8x + 4y - 5 = 0\) and \(S_1 = x^2 + y^2 + 2gx + 2fy = 0\), the common chord is found with

• \(S - S_1 = 0\)

This subtraction reveals the linear equation of the chord, which passes through the intersection points of the circles. By solving, the fixed point that the chord passes through is discovered, which often aligns with known conditions such as tangency or spatial constraints. Through intersections and elimination, the point is located based on geometric principles.

Tangent Line

In coordinate geometry, a tangent to a circle is a straight line that touches the circle at exactly one point. This creates a perpendicular relationship between the radius at that point and the tangent line. For the second circle in the problem, it touches the given line \(y=x\), indicating that line \(y=x\) is a tangent.
From this, the center of the circle can be understood to be equidistant from the line \(y=x\). This distance constraint assists in forming the general circle equation. Drawing from the property of a tangent, the point of tangency is significant for setting parameters (like \(g\) and \(f\) in the general circle equation) which affect both the circle's shape and position in the coordinate plane.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using the coordinate system. It connects algebraic equations with geometric figures, allowing us to analyze shapes as algebraic expressions like the circle equation. Here, equations yield detailed insights:

• Locations of points (e.g., centers, intersections).
• Relations between geometric entities (e.g., tangents).
• Curves, areas, and distances.

Crucially, it allows us to understand configurations like the common chord, calculating its exact position by separating equations of the intersecting circles. It reveals how lines, curves, and points fit together spatially, unlocking deeper intuitive and analytical understanding of geometry. This makes coordinate geometry an essential tool for dissecting problems into easier parts.