SATHEE: Circle 2 Question 21

Circle 2 Question 21

22. Let $C_{1}$ and $C_{2}$ be two circles with $C_{2}$ lying inside $C_{1}$ . A circle $C$ lying inside $C_{1}$ touches $C_{1}$ internally and $C_{2}$ externally. Identify the locus of the centre of $C$ .

$(2001, 5 M)$

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Solution:

Let the given circles $C_{1}$ and $C_{2}$ have centres $O_{1}$ and $O_{2}$ and radii $r_{1}$ and $r_{2}$ , respectively.

Let the variable circle $C$ touching $C_{1}$ internally, $C_{2}$ externally have a radius $r$ and centre at $O$ .

Now, $O O_{2} = r + r_{2}$ and $O O_{1} = r_{1} - r$

$\Rightarrow O O_{1} + O O_{2} = r_{1} + r_{2}$

which is greater than $O_{1} O_{2}$ as $O_{1} O_{2} < r_{1} + r_{2}$

$[∵ C_{2}$ lies inside $C_{1}]$

$\Rightarrow$ Locus of $O$ is an ellipse with foci $O_{1}$ and $O_{2}$ .

Alternate Solution

Let equations of $C_{1}$ be $x^{2} + y^{2} = r_{1}^{2}$ and of $C_{2}$ be $(x - a)^{2} + (y - b)^{2} = r_{2}^{2}$

Let cetnre $C$ be $(h, k)$ and radius $r$ , then by the given condition

$\begin{aligned} \sqrt{(h - a)^{2} + (k - b)^{2}} = r + r_{2} and \sqrt{h^{2} + k^{2}} & = r_{1} - r \\ \Rightarrow \sqrt{(h - a)^{2} + (k - b)^{2}} + \sqrt{h^{2} + k^{2}} & = r_{1} + r_{2} \end{aligned}$

Required locus is

$\sqrt{(x - a)^{2} + (y - b^{2})} + \sqrt{x^{2} + y^{2}} = r_{1} + r_{2}$

which represents an ellipse whose foci are $(a, b)$ and $(0, 0)$ .

Circle 2 Question 21

22. Let $C_{1}$ and $C_{2}$ be two circles with $C_{2}$ lying inside $C_{1}$ . A circle $C$ lying inside $C_{1}$ touches $C_{1}$ internally and $C_{2}$ externally. Identify the locus of the centre of $C$ .

Solution:

Alternate Solution

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Circle 2 Question 21

22. Let C1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1 touches C1 internally and C2 externally. Identify the locus of the centre of C.

Solution:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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22. Let $C_{1}$ and $C_{2}$ be two circles with $C_{2}$ lying inside $C_{1}$ . A circle $C$ lying inside $C_{1}$ touches $C_{1}$ internally and $C_{2}$ externally. Identify the locus of the centre of $C$ .