Problem 20 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) be the centres of the circles \(\mathrm{x}^{2}+y^{2}-2 \mathrm{x}-2 \mathrm{y}-2=0\) and \(x^{2}+y^{2}-6 x-6 y+14=0\) respectively. If \(P\) and \(Q\) are the points of intersection of these circles then, the area (in sq. units) of the quadrilateral \(\mathrm{PC}_{1} \mathrm{QC}_{2}\) is: (a) 8 (b) 6 (c) 9 (d) 4

Step-by-Step Solution

Answer

The area is 4 sq. units.

Step 1: Identify Centers of the Circles

First, rewrite the equations of the circles in standard form. The equation \(x^2 + y^2 - 2x - 2y - 2 = 0\) can be rewritten as \((x-1)^2 + (y-1)^2 = 4\). Thus, the center \(C_1\) is \((1, 1)\). For the second circle, \(x^2 + y^2 - 6x - 6y + 14 = 0\), it can be rewritten as \((x-3)^2 + (y-3)^2 = 4\), and the center \(C_2\) is \((3, 3)\).

Step 2: Determine Points of Intersection

The points of intersection \(P\) and \(Q\) of the circles can be determined by solving the two circle equations simultaneously. However, since the problem does not require exact coordinates for calculation of area, we will skip this explicit solving and understand that the centres are equidistant from \(P\) and \(Q\).

Step 3: Calculate Distance Between Centers

The distance \(C_1C_2\) between the centers \(C_1 = (1, 1)\) and \(C_2 = (3, 3)\) is given by the distance formula: \[\sqrt{(3-1)^2 + (3-1)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}.\]

Step 4: Apply Geometric Properties

In this configuration, \(P\) and \(Q\) are the endpoints of an imaginary line segment passing through the circle's centers because the circles intersect, and each intersection bisects this straight path into two equal parts.

Step 5: Compute the Area Using the Geometry of a Rhombus

Since the intersection is symmetrical and forms a perpendicular bisector, consider that the configuration forms a rhombus comprising the quadrilateral \(PC_1QC_2\). The side length of this rhombus is the radius of each circle which is 2. The area of a rhombus with side length \(a\) and angles of \(90^\circ\) is \(a^2 = 4.\) Therefore, the area of \(PC_1QC_2\) is 4 square units.

Key Concepts

Coordinates of circle centers

Finding the centers of circles defined by equations can be crucial in geometry, especially when dealing with problems involving intersecting circles. A standard form for a circle's equation is \[(x-h)^2 + (y-k)^2 = r^2,\] where

• \(h\) and \(k\) represent the \(x\) and \(y\) coordinates of the center respectively, and
• \(r\) is the radius of the circle.

To identify the center, you often need to rewrite the given circle equation into this standard form by completing the square. For instance, for the equation \(x^2 + y^2 - 2x - 2y - 2 = 0\), you rewrite it step-by-step to \((x-1)^2 + (y-1)^2 = 4\). This reveals that the center is at \((1, 1)\). Similarly, transforming \(x^2 + y^2 - 6x - 6y + 14 = 0\) to \((x-3)^2 + (y-3)^2 = 4\) shows that the center is at \((3, 3)\).
Understanding these transformations is fundamental when analyzing relationships between multiple circles like intersecting circles.

Distance formula

The distance formula is essential in geometry to measure the straight line distance between two points in a plane. The formula is stated as:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(D\) is the distance between these points.

For example, in our problem, the centers of the circles are \((1, 1)\) and \((3, 3)\). Plugging these into the formula gives us:\[D = \sqrt{(3 - 1)^2 + (3 - 1)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\]This concept is straightforward but important, as it forms the basis for determining how far apart points are in various geometric problems.

Intersecting circles

When two circles intersect, they share common points known as intersection points. In some problems, determining the exact location of these points may not be necessary if the geometric properties of the arrangement suffice.
For instance, if you know that the circles' centers are equally distant from these intersection points, it can help infer shape properties like symmetry. In the current problem, the focus is on the centers, rather than the exact coordinates of the intersection points \(P\) and \(Q\).
The configuration where the circles intersect points to a specific arrangement, typically symmetric across the line connecting the centers. This can often allow for the assumption of certain geometric properties, simplifying the calculation of areas or other measurements even without solving for \(P\) and \(Q\) specifically.

Area of quadrilateral

Calculating the area of quadrilaterals often requires knowledge of specific properties or dimensions. When dealing with a setup like the one with circles intersecting each other, understanding the geometric shape formed can simplify the process.
In this case, with the circles intersecting and organizing a quadrilateral such as \(PC_1QC_2\), knowing the nature of the quadrilateral can help. This particular arrangement suggests a rhombus-like shape, largely because the lines between intersection and center imply symmetrical distances and potentially right angles.
Given a known side length (here, the radius of the circles, \(2\)), the area of a simple figure like a rhombus can be determined using the formula:\[ \text{Area} = a^2 \]where \(a\) is the side length. Thus, knowing the radius is \(2\), the area calculates to \(2^2 = 4\) square units. This approach ties together understanding of symmetry, circle radii, and simple geometric formulas, providing a direct path to finding the area.