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A circle touches the x-axis and also touches the cirlce with centre at (0, 3) and radius 2. The locus of the centre of the cirlce is
- a)a circle
- b)an ellipse
- c)a parabola
- d)a hyperbola
Correct answer is option 'C'. Can you explain this answer?
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A circle touches the x-axis and also touches the cirlce with centre at...
Given data:
A circle touches the x-axis and also touches the circle with center at (0,3) and radius 2.
To find:
Locus of the center of the circle.
Solution:
Let the center of the required circle be (h,k) and its radius be r.
Now, we know that the given circle touches the x-axis.
So, the distance from the center of the circle to the x-axis will be equal to the radius of the circle.
Therefore, we have:
|k| = r
Also, the given circle touches the circle with center at (0,3) and radius 2.
So, the distance between the centers of the two circles will be equal to the sum of their radii.
Therefore, we have:
√((h-0)² + (k-3)²) = r+2
Squaring both sides, we get:
(h-0)² + (k-3)² = (r+2)²
Substituting |k| = r, we get:
(h-0)² + (r-3)² = (r+2)²
Expanding and simplifying, we get:
h² - 4r + 13 = 0
This is the equation of a parabola with vertex at (0,13/4) and axis of symmetry parallel to the x-axis.
Therefore, the locus of the center of the circle is a parabola.
Answer:
Option c) a parabola.
A circle touches the x-axis and also touches the circle with center at (0,3) and radius 2.
To find:
Locus of the center of the circle.
Solution:
Let the center of the required circle be (h,k) and its radius be r.
Now, we know that the given circle touches the x-axis.
So, the distance from the center of the circle to the x-axis will be equal to the radius of the circle.
Therefore, we have:
|k| = r
Also, the given circle touches the circle with center at (0,3) and radius 2.
So, the distance between the centers of the two circles will be equal to the sum of their radii.
Therefore, we have:
√((h-0)² + (k-3)²) = r+2
Squaring both sides, we get:
(h-0)² + (k-3)² = (r+2)²
Substituting |k| = r, we get:
(h-0)² + (r-3)² = (r+2)²
Expanding and simplifying, we get:
h² - 4r + 13 = 0
This is the equation of a parabola with vertex at (0,13/4) and axis of symmetry parallel to the x-axis.
Therefore, the locus of the center of the circle is a parabola.
Answer:
Option c) a parabola.
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A circle touches the x-axis and also touches the cirlce with centre at (0, 3) and radius 2. The locus of the centre of the cirlce isa)a circleb)an ellipsec)a parabolad)a hyperbolaCorrect answer is option 'C'. Can you explain this answer?
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