Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The locus of the mid-point of the chord of the circle \(x^{2}\) \(+y^{2}-2 x-2 y-2=0\) which makes an angle of \(120^{\circ}\) at the centre is (A) \(x^{2}+y^{2}-2 x-2 y+1=0\) (B) \(x^{2}+y^{2}+x+y-1=0\) (C) \(x^{2}+y^{2}-2 x-2 y-1=0\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A) \(x^2+y^2-2x-2y+1=0\).

Step 1: Rewrite the Circle Equation

The given circle equation is \(x^2 + y^2 - 2x - 2y - 2 = 0\). Rewrite it in standard form by completing the square.- \( (x^2 - 2x) + (y^2 - 2y) = 2 \)- Completing the square: \( (x-1)^2 - 1 + (y-1)^2 - 1 = 2 \)- Thus, \( (x-1)^2 + (y-1)^2 = 4 \)Now, the circle's center is at \((1,1)\) and the radius is \(2\).

Step 2: Understand Chord's Midpoint and Central Angle Condition

A chord of the circle subtends an angle of \(120^{\circ}\) at the center. For a given angle \( \theta \), the locus of the midpoint of the chord is another circle with the same center but a radius \( R \cos(\theta/2) \).

Step 3: Calculate Radius Using Angle Condition

For \(\theta = 120^{\circ}\), find the radius of the locus circle using the formula \( R \cos(\theta/2) \), where \(R = 2\) is the original circle's radius.- \(\cos(60^{\circ}) = \frac{1}{2}\)- Therefore, the radius of the new circle is \(2 \times \frac{1}{2} = 1\).

Step 4: Equation of Locus Circle

Since the locus is a circle with center \((1,1)\) and radius \(1\), we write the equation:- \( (x-1)^2 + (y-1)^2 = 1^2 \) - Expand the equation: \((x^2 - 2x + 1) + (y^2 - 2y + 1) = 1\)- Simplify to: \(x^2 + y^2 - 2x - 2y + 1 = 0\).

Step 5: Identify the Correct Answer

Compare the derived equation \(x^2 + y^2 - 2x - 2y + 1 = 0\) with the given options.- The correct option is (A) \(x^2 + y^2 - 2x - 2y + 1 = 0\).

Key Concepts

Circle Geometry

Circle geometry involves understanding the properties and equations of circles. A circle is a set of points that are equidistant from a fixed point called the center. This distance is known as the radius. The equation for a circle is typically written in the standard form

• \((x - h)^2 + (y - k)^2 = R^2\)

where

• \((h, k)\) are the coordinates of the center,
• and \(R\) is the radius.

In the exercise, we start with the expanded form of the circle's equation

• \(x^2 + y^2 - 2x - 2y - 2 = 0\).

To make it easier to understand the circle's properties, we need to convert this into its standard form. This involves completing the square, which helps us identify its center and radius clearly, finding that the center is \

• \((1,1)\)
• and the radius is \(2\).

Chord Properties

A chord is a line segment with both endpoints on the circle. The properties of chords, such as their length and the angles they form with the center, are crucial in circle geometry. For any given chord:

• The perpendicular from the center of the circle to the chord bisects the chord.
• The chord's midpoint lies equidistant from the center.

When a chord subtends a certain angle, in this case,

• \(120^{\circ}\) at the center, it provides information that helps in determining the locus of specific points related to the chord, such as its midpoint.

In our problem, understanding the angle subtended by the chord is crucial in finding the locus. The locus of points representing the midpoints of such chords forms another circle, which maintains certain proportional relationships with the original circle.

• The distance from the original circle's center to this locus is determined by the formula \( R \cos(\frac{\theta}{2})\).

Angle Subtended at the Center

The concept of an angle subtended at the center of a circle is important in determining geometric relationships within the circle. When a chord subtends an angle at the center, it establishes a distinct relationship between the circle and the chord.

• If a chord of a circle subtends an angle \(\theta\) at the center, the locus of the midpoint of this chord describes another circle.
• The center of this new circle is the same as the original circle.
• Its radius is computed using the formula \(R \cos(\frac{\theta}{2})\), where \(R\) is the radius of the original circle.

In the example provided, with a subtended angle of

• \(120^{\circ}\), the formula simplifies to calculating \(R \cos(60^{\circ})\), resulting in the new radius being half of \(R\), or \(1\).

The angle subtended consequently informs the size of the locus circle's radius and ensures it remains concentric with the original circle.