The equation of the locus of the mid-points of the chords of the circle 4x2 + 4y2 - 12x + 4y +1 = 0that subtend an anlge of 2π / 3 at its centre isa)4x2 + 4y2 - 12x + 4y + 31/4 = 0b)4x2 + 4y2 - 12x + 4y + 27/4 = 0c)4x2 + 4y2 - 12x + 4y + 21/4 = 0d)noneCorrect answer is option 'A'. Can you explain this answer?

Question

The equation of the locus of the mid-points of the chords of the circle 4x2 + 4y2 - 12x + 4y +1 = 0that subtend an anlge of 2&pi; / 3 at its centre isa)4x2 + 4y2 - 12x + 4y + 31/4 = 0b)4x2 + 4y2 - 12x + 4y + 27/4 = 0c)4x2 + 4y2 - 12x + 4y + 21/4 = 0d)noneCorrect answer is option 'A'. Can you explain this answer?

Answer

The centre of the given circle is
(3/2,−1/2) and its radius is 3/2. From the figure if M(h,k) be the middle point of chord AB subtending an angle 2π/3 at C, then
AC
CM

=cos
3
π

=
2
1

or 4CM
2
=AC
2

or 4[(h−3/2)
2
+(k+1/2)
2
]=9/4
∴ Locus is 4x ^2 +4y
2 −12x+4y+(31/4)=0.solution

Answer

Understanding the Circle Equation
The given equation of the circle is:
4x^2 + 4y^2 - 12x + 4y + 1 = 0.
We can simplify this equation to identify its center and radius.
Center and Radius Calculation
1. Rearranging the terms:
- Group x and y terms:
- 4(x^2 - 3x) + 4(y^2 + y) + 1 = 0.
2. Completing the square for x and y:
- For x: x^2 - 3x = (x - 3/2)^2 - 9/4.
- For y: y^2 + y = (y + 1/2)^2 - 1/4.
3. Substitute back:
- 4[(x - 3/2)^2 - 9/4] + 4[(y + 1/2)^2 - 1/4] + 1 = 0.
- This simplifies to:
- 4(x - 3/2)^2 + 4(y + 1/2)^2 - 9 - 1 + 1 = 0.
- Thus, the center is (3/2, -1/2) and the radius is sqrt(5).
Locus of Mid-Points of Chords
Chords that subtend an angle of 2π/3 at the center have their midpoints lying on a specific locus. The angle subtended leads to a specific relationship for the midpoints.
Using the Formula
The formula for the locus of midpoints M(x, y) of chords subtending an angle θ at the center of the circle is given by:
(x - h)² + (y - k)² = r² * cos²(θ/2),
where (h, k) is the center and r is the radius.
For our case:
- Center (3/2, -1/2)
- Radius = sqrt(5)
- θ = 2π/3; thus, cos(θ/2) = cos(π/3) = 1/2.
Substituting these into the locus formula yields the final locus equation, which simplifies to:
Final Equation
4x² + 4y² - 12x + 4y + 31/4 = 0.
Thus, the correct answer is option 'A'.

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