Circle 4 Question 1

1. The locus of the centres of the circles, which touch the circle, $x^{2}+y^{2}=1$ externally, also touch the $Y$-axis and lie in the first quadrant, is

(2019 Main, 10 April II)

(a) $y=\sqrt{1+2 x}, x \geq 0$

(b) $y=\sqrt{1+4 x}, x \geq 0$

(c) $x=\sqrt{1+2 y}, y \geq 0$

(d) $x=\sqrt{1+4 y}, y \geq 0$

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Answer:

Correct Answer: 1. (a)

Solution:

  1. Let $(h, k)$ be the centre of the circle and radius $r=h$, as circle touch the $Y$-axis and other circle $x^{2}+y^{2}=1$ whose centre $(0,0)$ and radius is 1.

$\therefore O C=r+1$

$[\because$ if circles touch each other externally, then $\left.C _1 C _2=r _1+r _2\right]$

$\Rightarrow \sqrt{h^{2}+k^{2}}=h+1, h>0$

and $k>0$, for first quadrant.

$\Rightarrow \quad h^{2}+k^{2}=h^{2}+2 h+1$

$\Rightarrow \quad k^{2}=2 h+1$

$\Rightarrow \quad k=\sqrt{1+2 h}$, as $k>0$

Now, on taking locus of centre $(h, k)$, we get

$y=\sqrt{1+2 x}, x \geq 0$



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