JAMIA 2021 — Mathematics PYQ

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Question

JAMIA 2021 — Mathematics PYQ

JAMIA | Mathematics | 2021

From the point A(3,2) a line is drawn to any point on the circle $x^{2} + y^{2} = 1.$ If locus of midpoint of this line seqment is a circle, then its radius is

Choose the correct answer:

Explanation

Given Information

Fixed Point: $A(3, 2)$
Given Circle: $x^2 + y^2 = 1$ (Radius $r = 1$ )
Variable Point on Circle: $P(h, k)$ where $h^2 + k^2 = 1$
Midpoint Locus: $M(X, Y)$

Locus Calculation

Using the midpoint formula for segment $AP$ :

X = \frac{h + 3}{2} \implies h = 2X - 3

Y = \frac{k + 2}{2} \implies k = 2Y - 2

Substitute $h$ and $k$ into $h^2 + k^2 = 1$ :

(2X - 3)^2 + (2Y - 2)^2 = 1

4\left(X - \frac{3}{2}\right)^2 + 4(Y - 1)^2 = 1

\left(X - \frac{3}{2}\right)^2 + (Y - 1)^2 = \frac{1}{4}

Final Answer

The equation represents a circle with:

Radius Squared: $R^2 = \frac{1}{4}$
$R^2 = \frac{1}{4}$