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A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]
- a)an ellipse
- b)a circle
- c)a hyperbola
- d)a parabola
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A circle touches the x- axis and also touches the circle with centre a...
Equation of circle with centre (0, 3) and radius 2 is x2 + (y - 3)2 = 4
Let locus of the variable circle is (α , β)
∵ It touches x - axis.
∴ It's equation is ( x - α)2 + (y + β)2 = β2
Circle touch externally ⇒ c1c2 = r1 + r2
∴ Locus is 
which is a parabola.
which is a parabola.Most Upvoted Answer
A circle touches the x- axis and also touches the circle with centre a...
The given situation:
- There is a circle with center at (0,3) and radius 2.
- Another circle touches the x-axis and also touches the circle with center at (0,3) and radius 2.
Understanding the problem:
- We need to find the locus of the center of the second circle.
Solution:
Step 1: Drawing the situation:
- Draw the circle with center at (0,3) and radius 2.
- Draw a tangent to this circle which touches the x-axis.
- Now draw the second circle which touches the x-axis and also touches the first circle.
Step 2: Analyzing the situation:
- Let the center of the second circle be (h, k).
- Since the second circle touches the x-axis, the distance between the center (h, k) and the x-axis is equal to the radius of the second circle.
- Since the second circle also touches the first circle, the distance between the centers of the two circles is equal to the sum of their radii.
Step 3: Using the distances:
- The distance between (h, k) and the x-axis is equal to the radius of the second circle, which is k.
- The distance between (h, k) and the center of the first circle (0,3) is equal to the sum of their radii, which is 2+k.
- So, we have two equations:
1. k = radius of the second circle
2. √(h²+(k-3)²) = 2+k
Step 4: Simplifying the equations:
- From equation (1), we have k = radius of the second circle.
- Substituting this value in equation (2), we get √(h²+(k-3)²) = 2+k.
- Squaring both sides of the equation, we get h²+(k-3)² = (2+k)².
- Expanding and simplifying the equation, we get h² + k² - 6k + 9 = k² + 4k + 4.
- Simplifying further, we get h² - 6k + 5 = 0.
Step 5: Finding the locus:
- The equation h² - 6k + 5 = 0 represents a parabola.
- Therefore, the locus of the center of the second circle is a parabola.
So, the correct answer is option 'D' - a parabola.
- There is a circle with center at (0,3) and radius 2.
- Another circle touches the x-axis and also touches the circle with center at (0,3) and radius 2.
Understanding the problem:
- We need to find the locus of the center of the second circle.
Solution:
Step 1: Drawing the situation:
- Draw the circle with center at (0,3) and radius 2.
- Draw a tangent to this circle which touches the x-axis.
- Now draw the second circle which touches the x-axis and also touches the first circle.
Step 2: Analyzing the situation:
- Let the center of the second circle be (h, k).
- Since the second circle touches the x-axis, the distance between the center (h, k) and the x-axis is equal to the radius of the second circle.
- Since the second circle also touches the first circle, the distance between the centers of the two circles is equal to the sum of their radii.
Step 3: Using the distances:
- The distance between (h, k) and the x-axis is equal to the radius of the second circle, which is k.
- The distance between (h, k) and the center of the first circle (0,3) is equal to the sum of their radii, which is 2+k.
- So, we have two equations:
1. k = radius of the second circle
2. √(h²+(k-3)²) = 2+k
Step 4: Simplifying the equations:
- From equation (1), we have k = radius of the second circle.
- Substituting this value in equation (2), we get √(h²+(k-3)²) = 2+k.
- Squaring both sides of the equation, we get h²+(k-3)² = (2+k)².
- Expanding and simplifying the equation, we get h² + k² - 6k + 9 = k² + 4k + 4.
- Simplifying further, we get h² - 6k + 5 = 0.
Step 5: Finding the locus:
- The equation h² - 6k + 5 = 0 represents a parabola.
- Therefore, the locus of the center of the second circle is a parabola.
So, the correct answer is option 'D' - a parabola.
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A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer?
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A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer?.
A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer?.
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A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of A circle touches the x- axis and also touches the circle with centre at (0,3 ) and radius 2. The locus of the centre of the circle is[2005]a)an ellipseb)a circlec)a hyperbolad)a parabolaCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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