NIMCET 2023 — Mathematics PYQ

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Question

NIMCET 2023 — Mathematics PYQ

NIMCET | Mathematics | 2023

A circle touches the X -axis and also touches the circle with centre (0,3) and radius 2. The locus of the centre of the circle is

Choose the correct answer:

Explanation

CONCEPT:
When two circles having radius $r_{1}$ and $r_{2}$ and centers at $C_{1}$ and $C_{2}$ respective touches each other then $∣ C_{1} C_{2} ∣ = r_{1} + r_{2}$

CALCULATION:
Let $C_{1} (h, k)$ be the center of the circle.
$∴$ Circle touches the x-axis then its radius is $r_{1} = k$ .

Also, the circle with centre $C_{1}$ touches the another circle with centre $C_{2} (0, 3)$ and radius $r_{2} = 2$ .

$∴ ∣ C_{1} C_{2} ∣ = r_{1} + r_{2}$
$< b r >\Rightarrow (h - 0)^{2} + (k - 3)^{2} = ∣ k + 2∣ < b r >$
On Squaring both the sides of the above equation we get
$< b r >\Rightarrow h^{2} - 10 k + 5 = 0 < b r >$
Now replace h with x and k with y in the above equation we get
$\Rightarrow$ Locus is $x^{2} - 10 y + 5 = 0$ , which is parabola.

Therefore, option (1) is the correct answer.

Choose the correct answer:

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