Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1: The only circle having radius \(\sqrt{10}\) and a diameter along line \(2 x+y=5\) is \(x^{2}+y^{2}-6 x+2 y=0\). Statement \(\mathbf{2}: 2 x+y=5\) is a normal to the circle $x^{2}+y^{2}-6 x+2 y=0 . (a) Statement 1 is false; Statement 2 is true. (b) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1 . (c) Statement 1 is true; Statement 2 is false. (d) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 .

Step-by-Step Solution

Answer

(d) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

Step 1: Identify the Circle Equation Features

Given the circle equation \(x^2 + y^2 - 6x + 2y = 0\), we start by comparing it with the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\). Expanding and comparing terms, we identify that \(2gh = -6\) and \(2fk = 2\). Thus, the center \((h,k)\) of the circle is \((3,-1)\). The standard form can be written as \((x-3)^2 + (y+1)^2 = r^2\).

Step 2: Calculate the Circle's Radius

To find \(r\) for the circle, we proceed using \(r^2\) equals the constant on the right side of the circle equation. We complete the square to find \((x-3)^2 + (y+1)^2 = 10\), so \(r^2 = 10\), this gives us \(r = \sqrt{10}\).

Step 3: Verify if Diameter Lies on the Line

In statement 1, the diameter is said to lie along the line \(2x+y=5\). For a line to be a diameter, the center should lie on this line. Substitute the circle's center \((3,-1)\) into the line equation: \(2(3) + (-1) = 5\), which holds true. Therefore, statement 1 is correct.

Step 4: Check if the Line is Normal to the Circle

Statement 2 claims that the line \(2x+y=5\) is a normal to the circle. First, find the slope of the line, which is \(-2\). The slope of the radius (which is also the slope from the circle center to the line) is \(-\frac{1}{2}\) since the line \(2x+y=5\) is perpendicular to the segment joining the center. Since this perpendicular slope is observed, the line being a normal to the circle is true, making statement 2 correct.

Step 5: Determine Relationship Between Statements

Since the diameter coinciding with the line and the normal claim for the circle are independently true but unrelated in the context of providing an explanation, statement 2 doesn't explain statement 1.

Key Concepts

Circle Equations

Circle equations allow us to express the properties of a circle using algebraic terms. A standard circle equation in coordinate geometry looks like this:

• \((x-h)^2 + (y-k)^2 = r^2\)

In this formula,

• \( (h, k) \) denotes the center of the circle,
• \( r \) stands for the radius of the circle.

The expression expands to form equations that involve terms of \(x^2\), \(y^2\), \(x\), and \(y\), while comparing with non-standard forms is useful for identifying the circle's properties.
To identify the center and radius from a given circle equation, it's crucial to complete the square for both \(x\) and \(y\) terms. This method reveals the geometric characteristics of the circle, making circle equations a fundamental tool for solving geometry problems.

Circle Radius

The circle radius is the distance from the center of the circle to any point on its circumference. It is a constant value that defines the size of the circle. In the equation

• \((x-h)^2 + (y-k)^2 = r^2\),

\( r \) denotes the radius.
Finding the radius involves determining this value, which is usually achieved by solving for \(r\) in the equation:

• The radius \(r\) is the square root of the constant term on the right side of the standard circle form equation. For example, if the equation resembles
  • \((x-3)^2 + (y+1)^2 = 10\),
  then
  • \(r^2 = 10\),
  • thus \(r = \sqrt{10}\).

Therefore, calculating the circle radius accurately interprets how expansive the circle is in relation to its center.

Normal to a Circle

Understanding the concept of a "normal" to a circle involves recognizing the perpendicular line from a given point to the circle's curve. In geometric terms, a normal is a line that forms a 90-degree angle with the tangent of the circle at its point of contact.
For practical identification, a line's slope is crucial:

• If a line is normal to a circle, its slope must be perpendicular to the radius of the circle at the point of contact.

For example, suppose a line has a slope of

• \(-2\) and is normal to the circle,

then the radius joining the center of the circle to the point of contact would have a slope of the negative reciprocal,

• \(-\frac{1}{2}\).

This property ensures that the line properly serves as a normal, emphasizing the relationship between a circle's center and its tangential points.

Diameter and Chord

A diameter is a special type of chord that passes through the center of a circle, making it the longest possible chord of the circle. To understand this practically, consider the line segment extending from one point on the circle's circumference, passing through the center, to another point.
To find if a line segment is a diameter:

• Ensure that the midpoint of the segment coincides with the center of the circle.

The diameter's key feature is that it divides the circle into two equal halves.
A regular chord is a line joining any two points on the circle and does not necessarily pass through the center.
This understanding of a diameter and chord is fundamental in circle geometry, as it relates to other properties like area and circumference and provides insight into additional geometric constructions.