Problem 112 - A Complete Resource Book in Mathematics for JEE Main

Question

If the two circles \((x-1)^{2}+(y-3)^{2}=r^{2}\) and \(x^{2}+y^{2}\) \(-8 x+2 y+8=0\) intersect in two distinct points, then [2003] (A) \(22\)

Step-by-Step Solution

Answer

The answer is (A) \(2 < r < 8\).

Step 1: Identify the Circle Equations

First, identify and understand the equations of the two given circles. The first circle is \((x-1)^2 + (y-3)^2 = r^2\). The second circle can be rewritten from \(x^2 + y^2 - 8x + 2y + 8 = 0\) to \((x-4)^2 + (y+1)^2 = 9\) by completing the square.

Step 2: Calculate the Distance Between Centers

Next, calculate the distance between the centers of the two circles. The center of the first circle is \((1, 3)\) and the center of the second circle is \((4, -1)\). The distance between them is \(d = \sqrt{(4-1)^2 + (-1-3)^2} = \sqrt{9 + 16} = 5\).

Step 3: Apply Conditions for Intersecting Circles

For two circles to intersect at exactly two points, the distance between their centers \(d\) must be less than the sum of their radii and greater than the absolute difference of their radii. In this case, the conditions are: \(|r - 3| < 5 < r + 3\) since the radius of the second circle is 3.

Step 4: Solve Inequalities

Solve the inequalities \(|r - 3| < 5\) and \(r + 3 > 5\). \(|r - 3| < 5\) gives \(-5 < r - 3 < 5\), which simplifies to \(-2 < r < 8\). The inequality \(r + 3 > 5\) simplifies to \(r > 2\).

Step 5: Determine the Correct Range for r

Combine the inequalities from Step 4: \(r > 2\) and \(-2 < r < 8\), resulting in \(2 < r < 8\). Because the inequality \(r > 2\) concurs with \(-2 < r\), the valid range for \(r\) that allows for two points of intersection is \(2 < r < 8\).

Key Concepts

Equation of a Circle

The equation of a circle is a fundamental concept in geometry and represents all the points that are equidistant from a fixed center point in a plane. Typically, a circle is described by the equation \[(x - h)^2 + (y - k)^2 = r^2\]where

• \((h, k)\) is the center of the circle,
• \(r\) is the radius of the circle.

Let's consider the first circle given in the exercise: \((x-1)^2 + (y-3)^2 = r^2\).Here, the center is \((1, 3)\) and the radius is \(r\). For the second circle, \(x^2 + y^2 - 8x + 2y + 8 = 0\), first, complete the square to rewrite it in the standard form \((x-4)^2 + (y+1)^2 = 9\).Thus, the center is \((4, -1)\) and the radius is \(3\). Understanding these equations is essential to determine how these circles might intersect.

Distance Between Centers

The distance between the centers of two circles is crucial in determining how they interact with each other geometrically. To find this distance, use the distance formula based on the coordinates of the circle centers: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]For the circles in our problem, locate the centers as \((1, 3)\) and \((4, -1)\). Then, substitute these into the distance formula: \[d = \sqrt{(4-1)^2 + (-1-3)^2} = \sqrt{9 + 16} = 5\].This calculation shows the distance between the centers is \(5\). This distance helps determine if, and at how many points, the circles might intersect, by comparing it to their radii.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry through graphs and equations. It allows us to explore geometric problems using coordinate systems like the cartesian coordinate plane. In this exercise, it plays a key role in analyzing the position and interaction of the two circles.
The equations of the circles are written in the coordinate plane, enabling us to graph them and analyze their properties, such as centers and radii. Moreover, coordinate geometry allows us to use powerful tools like the distance formula and solving inequalities involving circle radii to identify intersections.

• The cartesian plane provides a visual and measurable way to understand how circular equations manifest geometrically.
• It also allows for algebraic manipulation (completing the square, solving equations) to extract meaningful geometric properties.

Thus, coordinate geometry is invaluable for discerning the specific intersection conditions of the circles.

Radii Conditions for Intersection

For two circles to intersect at two distinct points, certain conditions about their radii and the distance between centers must be satisfied. These conditions are derived from the geometric properties of circles.
Firstly, the sum of their radii must be greater than the distance between their centers, \(r_1 + r_2 > d\).Simultaneously, the absolute difference of their radii must be less than this center distance, \(|r_1 - r_2| < d\).In our exercise, the second circle's radius \(r_2\) is \(3\), and the center distance is \(5\). Therefore, the conditions become:

• \(|r - 3| < 5\)
• \(r + 3 > 5\)

By solving these inequalities,

• the first inequality, \(|r - 3| < 5\), simplifies to \(-2 < r < 8\),
• while the second imposes \(r > 2\).

Combining these, the viable range for \(r\) that allows exactly two intersection points is \(2 < r < 8\).These conditions ensure the circles are close enough to intersect without being entirely separate or having one circle within the other.