SATHEE: Circle 4 Question 1

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

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

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Answer:

Correct Answer: 1. (a)

Solution:

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.

$∴ O C = r + 1$

$[∵$ if circles touch each other externally, then $C_{1} C_{2} = r_{1} + r_{2}]$

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

and $k > 0$ , for first quadrant.

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

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

$\Rightarrow k = \sqrt{1 + 2 h}$ , 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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Circle 4 Question 1

1. The locus of the centres of the circles, which touch the circle, x2+y2=1 externally, also touch the Y-axis and lie in the first quadrant, is

Answer:

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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