
The locus of the center of a circle which touches the circles and externally ( are complex numbers) will be-
Answer
459.9k+ views
Hint: Here we will first draw the diagram of the circles which touch the other two circles externally. Then we will see the distance between their centers and from those equations we will find the other equation. Then we will see or check whether the final equation satisfies any of the curve property.
Complete Complete Step by Step Solution:
It is given that the circle touches the other two circle i.e. and externally.
So, the center of the circle one is with radius and the center of the other circle is with radius .
Now we will draw these two circles with the third circles which touch them externally. So, we get
Now we know that the distance between the centers of the circle is equal to the sum of their radius. Therefore, we can write it as
…………….
Similarly writing the distance equation for other circle, we get
…………….
Now from the equation we will find the value of . So, we get
Substituting the value of in the equation , we get
Now subtracting from both the sides, we get
Now, from equation , we get
Substituting this value in equation , we get
Now subtracting from both the sides, we get
Now from both the equations, we got the same result.
This means that the difference between the distances of the center of the circles is constant.
As we know that the distance between the foci of the hyperbola remains constant i.e. .
Then by comparing this with the obtained equation, we can say that the locus of the center of a circle which touches the circles and externally will be a Hyperbola.
Hence, the locus of the center of a circle which touches the circles externally will be a Hyperbola.
Note:
1) Here we should know that the distance between the foci of the hyperbola always remains constant i.e. also the distance between the centers of the two external touching circles is equal to the sum of their radius.
2) While solving this question we have to make an equation that satisfies the property of the curve then we can say that the locus of the center of a circle is of that type of curve.
Complete Complete Step by Step Solution:
It is given that the circle touches the other two circle i.e.
So, the center of the circle one is
Now we will draw these two circles with the third circles which touch them externally. So, we get

Now we know that the distance between the centers of the circle is equal to the sum of their radius. Therefore, we can write it as
Similarly writing the distance equation for other circle, we get
Now from the equation
Substituting the value of
Now subtracting
Now, from equation
Substituting this value in equation
Now subtracting
Now from both the equations, we got the same result.
This means that the difference between the distances of the center of the circles is constant.
As we know that the distance between the foci of the hyperbola remains constant i.e.
Then by comparing this with the obtained equation, we can say that the locus of the center of a circle which touches the circles
Hence, the locus of the center of a circle which touches the circles externally will be a Hyperbola.
Note:
1) Here we should know that the distance between the foci of the hyperbola always remains constant i.e.
2) While solving this question we have to make an equation that satisfies the property of the curve then we can say that the locus of the center of a circle is of that type of curve.
Latest Vedantu courses for you
Grade 11 Science PCM | CBSE | SCHOOL | English
CBSE (2025-26)
School Full course for CBSE students
₹41,848 per year
Recently Updated Pages
Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

How do you convert 295 meters per second to kilometers class 7 maths CBSE

Write the following in Roman numerals 25819 class 7 maths CBSE

Trending doubts
State and prove Bernoullis theorem class 11 physics CBSE

What are Quantum numbers Explain the quantum number class 11 chemistry CBSE

Write the differences between monocot plants and dicot class 11 biology CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

State the laws of reflection of light

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
