GATE 2025 PH – Question 9

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

A circle with center at (x, y) = (0.5, 0) and radius = 0.5 intersects with another circle with center at (x, y) = (1, 1) and radius = 1 at two points. One of the points of intersection (x, y) is:

(0, 0)
(0.2, 0.4)
(0.5, 0.5)
(1, 2)

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Correct answer: (B) (0.2, 0.4)

Explanation

The two circles are
$$C_1:\ (x-0.5)^2+y^2=0.25,\qquad C_2:\ (x-1)^2+(y-1)^2=1 .$$

A point of intersection must satisfy **both** equations. Test the options.

**(0, 0):** $C_1$: $0.25+0=0.25$ ✓. $C_2$: $1+1=2\neq1$ ✗.
**(0.2, 0.4):** $C_1$: $(0.2-0.5)^2+0.4^2=0.09+0.16=0.25$ ✓. $C_2$: $(0.2-1)^2+(0.4-1)^2=0.64+0.36=1$ ✓. **Both satisfied.**
**(0.5, 0.5):** $C_1$: $0+0.25=0.25$ ✓. $C_2$: $0.25+0.25=0.5\neq1$ ✗.
**(1, 2):** $C_1$: $0.25+4\neq0.25$ ✗.

The point of intersection is **(0.2, 0.4)** (option B).