Problem 66 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The number of common tangents of the circles given by \(x^{2}+y^{2}-8 x-2 y+1=0\) and \(x^{2}+y^{2}+6 x+8 y=0\) is (a) one (b) four (c) two (d) three

Step-by-Step Solution

Answer

There are two common tangents.

Step 1: Find the Centers and Radii of the Circles

First, rewrite each circle equation in standard form to identify their centers and radii. For the first circle, the equation is \( x^2 + y^2 - 8x - 2y + 1 = 0 \). Completing the square gives:\[(x-4)^2 + (y-1)^2 = 16\].This circle has a center at \((4, 1)\) and a radius \(r_1 = 4\).For the second circle, the equation is \( x^2 + y^2 + 6x + 8y = 0 \). Completing the square gives:\[(x+3)^2 + (y+4)^2 = 25\].This circle has a center at \((-3, -4)\) and a radius \(r_2 = 5\).

Step 2: Calculate the Distance Between the Centers

Calculate the distance \(d\) between the two centers using the distance formula:\[d = \sqrt{(4 - (-3))^2 + (1 - (-4))^2} = \sqrt{(4+3)^2 + (1+4)^2} = \sqrt{49 + 25} = \sqrt{74}\].The distance between the centers is \(d = \sqrt{74}\).

Step 3: Evaluate the Conditions for Common Tangents

For two circles, the number of common tangents is determined based on their radii \(r_1\), \(r_2\), and the distance \(d\) between their centers:- If \(d > r_1 + r_2\), there are 4 common tangents.- If \(d = r_1 + r_2\), there are 3 common tangents.- If \(d < r_1 + r_2\) and \(d > |r_1 - r_2|\), there are 2 common tangents.- If \(d = |r_1 - r_2|\), there is 1 common tangent.- If \(d < |r_1 - r_2|\), there are no common tangents. In this case, \(r_1 = 4\), \(r_2 = 5\), and \(d = \sqrt{74}\). Calculate:- \(r_1 + r_2 = 9\)- \(|r_1 - r_2| = 1\).

Step 4: Determine the Number of Common Tangents

Since \(|r_1 - r_2| = 1 < \sqrt{74} < 9 = r_1 + r_2\), the condition \(d < r_1 + r_2\) and \(d > |r_1 - r_2|\) holds. Hence, there are 2 common tangents.

Key Concepts

Circle Equations

Circle equations provide the mathematical way of representing a circle on a coordinate plane. A circle, defined in standard form, uses the formula \[(x-h)^2 + (y-k)^2 = r^2,\]where

• \((h, k)\) represents the center of the circle, providing the coordinates where the circle is centered.
• \(r\) stands for the radius of the circle, the distance from the center to any point on the boundary of the circle.

To convert a general circle equation such as \(x^2 + y^2 + ax + by + c = 0\) into its standard form, you need to use a method called completing the square. This allows you to clearly see the center and radius of the circle, making it easier to work with in various mathematical applications. Completing the square involves rearranging and reframing the equation so it fits the form above, indicating the circle's precise attributes. This transformation is key for further calculations, such as determining properties like common tangents.

Distance Between Centers

Understanding the distance between the centers of two circles is crucial for many calculations, including determining common tangents. This distance is found using the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\]where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the centers of the two circles.
• \(d\) represents the distance between these two points.

Calculating this distance helps in understanding how two circles relate to one another in a plane. Whether they are intersecting, touching externally or internally, or completely apart influences the number of potential common tangents they can have. Hence, recognizing this distance forms a pivotal part of understanding and solving problems associated with circle configurations.

Radii of Circles

The radius of a circle is a fundamental concept that defines the circle's size and is half the diameter. It is the length from the center of the circle to any point on its edge. Calculating and understanding the radius is essential, particularly when addressing problems like common tangents or circle intersections.Given the circle's standard equation \[(x-h)^2 + (y-k)^2 = r^2,\] \(r\) is derived directly as the square root of the right side. Knowing the radius helps in deducing various circle properties, such as

• how far the circle extends on the coordinate plane,
• calculating areas and circumferences,
• determining overlap with other circles,
• and most importantly, how circles can interact based on their relative sizes and positions.

The radius serves as a vital tool in pinpointing these relationships, playing a crucial role in understanding circle geometry.