Problem 32 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A circle passes through the points \((2,3)\) and \((4,5) .\) If its centre lies on the line, \(y-4 x+3=0\), then its radius is equal to (a) \(\sqrt{5}\) (b) 1 (c) \(\sqrt{2}\) (d) 2

Step-by-Step Solution

Answer

The radius of the circle is 2.

Step 1: Understand the Problem

We need to find the radius of a circle that passes through two points, \((2,3)\) and \((4,5)\), and whose center lies on the line \(y - 4x + 3 = 0\). We will solve this by finding the equation for the center of the circle and then using any of the given points to calculate the radius.

Step 2: Equation for Circle Center

The equation for the center \((h, k)\) of the circle must satisfy the line equation \(y - 4x + 3 = 0\). Thus, \(k - 4h + 3 = 0\) gives \(k = 4h - 3\).

Step 3: Distance from Center to Points

The circle passes through \((2,3)\) and \((4,5)\), so the distance from the center \((h, k)\) to these points will be the radius \(r\). We first calculate the distance between \((h, k)\) and \((2, 3)\). The distance formula gives \(r = \sqrt{(h - 2)^2 + (k - 3)^2}\).

Step 4: Substitute Center Equation

Replace \(k\) in the distance formula with \(4h - 3\) from the line equation. Thus,\[r = \sqrt{(h - 2)^2 + ((4h - 3) - 3)^2} = \sqrt{(h - 2)^2 + (4h - 6)^2}\].

Step 5: Set Equation for Second Point

Use the same distance relationship for point \((4, 5)\): Set \(r = \sqrt{(h - 4)^2 + (k - 5)^2}\). Substitute \(k = 4h - 3\):\[r = \sqrt{(h - 4)^2 + ((4h - 3) - 5)^2} = \sqrt{(h - 4)^2 + (4h - 8)^2}\].

Step 6: Equate Distances

Since both distances \(r\) are equal (as they are the same radius), equate:\[(h - 2)^2 + (4h - 6)^2 = (h - 4)^2 + (4h - 8)^2\].

Step 7: Simplify and Solve for h

Simplify both sides and solve for \(h\):Expanding:\[(h^2 - 4h + 4) + (16h^2 - 48h + 36) = (h^2 - 8h + 16) + (16h^2 - 64h + 64)\].Combine like terms and solve:\[17h^2 - 52h + 40 = 17h^2 - 72h + 80\].This simplifies to \(20h = 40\), giving \(h = 2\).

Step 8: Find k Using h

With \(h = 2\), substitute back into \(k = 4h - 3\):\[k = 4(2) - 3 = 8 - 3 = 5\].Thus, the center is \((2, 5)\).

Step 9: Calculate Radius

Using the center \((2, 5)\) and one of the given points, \((2, 3)\): The distance, which is the radius, is\[r = \sqrt{(2 - 2)^2 + (5 - 3)^2} = \sqrt{0 + 4} = 2\].

Step 10: Determine Answer Choice

The calculated radius is 2. Hence, the correct answer choice is option (d).

Key Concepts

Circle Radius

Understanding the circle radius is crucial in solving problems involving circles in coordinate geometry. The radius of a circle is the distance from the circle's center to any point on its circumference. For our problem, the circle passes through points

• (2, 3)
• (4, 5)

and its center lies on a specific line. To find the radius, we need the distance from the center, once located, to one of these points. The standard approach is using the distance formula, making calculation straightforward once the center is determined.

Distance Formula

The distance formula is a valuable tool in geometry, especially in calculating the radius of a circle on a plane. We use the formula to measure the distance between two points, which is the key to finding the radius. The formula is written as: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
In our exercise, the distance formula helps calculate the radius by finding the distance between the center of the circle and the points through which the circle passes. Once the coordinates of the center are identified, simply plug them into the formula with one of the given points and solve.

Equation of a Line

The equation of a line is essential in this problem because the center of the circle lies on this line. For the line equation given as\[y - 4x + 3 = 0\],we derive the relationship between x and y-coordinates of any point on the line. Specifically, it allows us to express

• \(k = 4h - 3\)

for the center coordinates

• (h, k)

This equation is useful in substituting back into the distance formula, ensuring that the center calculated lies correctly on the line described, leading to finding the accurate radius of the circle.

Coordinate Geometry

Coordinate geometry merges algebra and geometry, making it simpler to solve complex geometric problems such as those involving circles. In this exercise, we integrate various geometric concepts to arrive at the solution. We are given two points and a line.

• First, identify the line equation.
• Use calculations to identify possible centers of the circle along this line.
• Apply the distance formula to these points to find possible radii.

The power of coordinate geometry lies in its ability to break down problems into smaller steps, like this. It allows us to combine equations and calculations to find a singular solution. This intertwining methodology leads us from understanding the problem to finding a precise answer, as we did for finding the radius.