NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

A circle touches the X-axis and also touches another circle with centre at $(0, 3)$ and radius $2$ . Then the locus of the centre of the first circle is:

Choose the correct answer:

Explanation

1. Condition for touching the X-axis:

Since the circle touches the X-axis, its radius must be equal to the absolute value of its y-coordinate.

r = |k|

2. Condition for touching the second circle:

The second circle has centre $C_2(0, 3)$ and radius $r_2 = 2$ .

When two circles touch each other (let's assume they touch externally for the general locus), the distance between their centres is equal to the sum of their radii:

C_1C_2 = r + r_2

3. Applying the Distance Formula:

The distance between $C_1(h, k)$ and $C_2(0, 3)$ is:

\sqrt{(h - 0)^2 + (k - 3)^2} = |k| + 2

4. Squaring both sides:

(h - 0)^2 + (k - 3)^2 = (|k| + 2)^2

h^2 + k^2 - 6k + 9 = k^2 + 4|k| + 4

Assuming the circle is above the X-axis ( $k > 0$ ), we replace $|k|$ with $k$ :

h^2 + k^2 - 6k + 9 = k^2 + 4k + 4

h^2 - 10k + 5 = 0

h^2 = 10k - 5

5. Identifying the Locus:

To find the locus, we replace $(h, k)$ with $(x, y)$ :

x^2 = 10y - 5

This equation is of the form $x^2 = 4ay + c$ , which represents a parabola.

Correct Option:

(a) a parabola