Let C be the circle with centre (0,0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle c that subtend an angle of at its centre is - [AIEEE-2006]a)x2+ y2= 1b)x2+ y2= 27/4c)x2+ y2= 9/4d)x2+ y2= 3/2Correct answer is option 'C'. Can you explain this answer?

Question

Let C be the circle with centre (0,0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle c that subtend an angle of at its centre is -   [AIEEE-2006]a)x2+ y2= 1b)x2+ y2= 27/4c)x2+ y2= 9/4d)x2+ y2= 3/2Correct answer is option 'C'. Can you explain this answer?

Answer

An angle of 90 degrees???

Answer

To find the locus of the midpoints of the chords of the given circle, we need to consider the chords that subtend a particular angle at the center.

Let's consider a chord AB that subtends an angle θ at the center of the circle.

Finding the Midpoint:
To find the midpoint of the chord AB, we need to find the average of the x-coordinates and the average of the y-coordinates of points A and B.

Let A and B have coordinates (x1, y1) and (x2, y2) respectively.

The midpoint M will have coordinates:
x = (x1 + x2)/2
y = (y1 + y2)/2

Using the properties of chords subtending an angle θ:
For a given angle θ, the coordinates of points A and B can be represented as:
A = (3cos(θ/2), 3sin(θ/2))
B = (3cos(θ/2 + θ), 3sin(θ/2 + θ))

Substituting these coordinates into the midpoint formulas, we get:
x = (3cos(θ/2) + 3cos(θ/2 + θ))/2
y = (3sin(θ/2) + 3sin(θ/2 + θ))/2

Simplifying the expressions:
x = (3cos(θ/2) + 3cos(θ/2)cos(θ) - 3sin(θ/2)sin(θ))/2
y = (3sin(θ/2) + 3sin(θ/2)cos(θ) + 3cos(θ/2)sin(θ))/2

Using trigonometric identities:
cos(θ/2)cos(θ) - sin(θ/2)sin(θ) = cos(θ/2 + θ)
sin(θ/2)cos(θ) + cos(θ/2)sin(θ) = sin(θ/2 + θ)

Simplifying further:
x = (3cos(θ/2 + θ))/2
y = (3sin(θ/2 + θ))/2

The locus of the midpoints can be represented by the parametric equations:
x = (3cos(θ/2 + θ))/2
y = (3sin(θ/2 + θ))/2

Eliminating the parameter θ:
x^2 = (9cos^2(θ/2 + θ))/4
y^2 = (9sin^2(θ/2 + θ))/4

Using the trigonometric identity:
cos^2(θ/2 + θ) + sin^2(θ/2 + θ) = 1

We get:
x^2 + y^2 = 9/4

Thus, the equation of the locus of the midpoints of the chords of the given circle is:
x^2 + y^2 = 9/4, which corresponds to option C.

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