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.
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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.
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