SATHEE: Circle 4 Question 7

Circle 4 Question 7

7. The locus of the centre of circle which touches $(y - 1)^{2} + x^{2} = 1$ externally and also touches $X$ -axis, is

(a) $x^{2} = 4 y, y \geq 0 \cup (0, y), y < 0$

(b) $x^{2} = y$

(d) $y^{2} = 4 x \cup (0, y), y \in R$

(2005, 2M)

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Answer:

Correct Answer: 7. (a)

Solution:

Let the locus of centre of circle be $(h, k)$ touching $(y - 1)^{2} + x^{2} = 1$ and $X$ -axis shown as

Clearly, from figure,

Distance between $C$ and $A$ is always $1 + | k |$ ,

i.e. $\sqrt{(h - 0)^{2} + (k - 1)^{2}} = 1 + | k |$ ,

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

$\Rightarrow h^{2} = 2 | k | + 2 k \Rightarrow x^{2} = 2 | y | + 2 y$

where $| y | = \begin{matrix} y, y \geq 0 - y, y < 0 \end{matrix}$

$∴ x^{2} = 2 y + 2 y, y \geq 0$

and $x^{2} = - 2 y + 2 y, y < 0$

$\Rightarrow x^{2} = 4 y$ , when $y \geq 0$

and $x^{2} = 0$ , when $y < 0$

$∴ (x, y) : x^{2} = 4 y .$ , when $. y \geq 0 \cup (0, y) : y < 0$

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Circle 4 Question 7

7. The locus of the centre of circle which touches (y−1)2+x2=1 externally and also touches X-axis, is

Answer:

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7. The locus of the centre of circle which touches $(y - 1)^{2} + x^{2} = 1$ externally and also touches $X$ -axis, is