Question:
A circle touches the x-axis and also touches the circle which centre at \( (0, 3) \) and radius 2. The locus of the center of the circle is
A circle touches the x-axis and also touches the circle which centre at \( (0, 3) \) and radius 2. The locus of the center of the circle is
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When solving locus problems involving tangency conditions, use the distance formula to relate the distances between the centers of the circles and apply the tangency condition.
Updated On: Apr 22, 2026
- a parabola
- a circle
- an ellipse
- a hyperbola
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The Correct Option is A
Solution and Explanation
Step 1: Understand the geometry of the problem.
We are given a circle that touches the x-axis and also touches another circle with center at \( (0, 3) \) and radius 2. We need to find the locus of the center of the circle.
Step 2: Analyze the situation geometrically.
The first circle touches the x-axis, meaning the distance from its center to the x-axis is equal to its radius, say \( r \). The second circle has a center at \( (0, 3) \) and radius 2.
Step 3: Use the distance condition.
The distance between the centers of the two circles is equal to the sum of their radii, i.e., the distance between the center of the first circle and \( (0, 3) \) is \( r + 2 \).
Step 4: Equation of the locus.
Let the center of the first circle be \( (x, y) \). The distance between \( (x, y) \) and \( (0, 3) \) is given by the equation: \[ \sqrt{x^2 + (y - 3)^2} = r + 2 \]
Step 5: Use the condition that the circle touches the x-axis.
Since the circle touches the x-axis, the distance from its center to the x-axis is equal to its radius \( r \). Therefore, the equation becomes: \[ y = r \]
Step 6: Conclusion.
By substituting \( y = r \) into the distance equation, we obtain the equation of a parabola. Therefore, the locus of the center of the circle is a parabola, corresponding to option (A).
Step 7: Verification.
By analyzing the geometry and applying the distance conditions, we confirm that the locus is indeed a parabola.
We are given a circle that touches the x-axis and also touches another circle with center at \( (0, 3) \) and radius 2. We need to find the locus of the center of the circle.
Step 2: Analyze the situation geometrically.
The first circle touches the x-axis, meaning the distance from its center to the x-axis is equal to its radius, say \( r \). The second circle has a center at \( (0, 3) \) and radius 2.
Step 3: Use the distance condition.
The distance between the centers of the two circles is equal to the sum of their radii, i.e., the distance between the center of the first circle and \( (0, 3) \) is \( r + 2 \).
Step 4: Equation of the locus.
Let the center of the first circle be \( (x, y) \). The distance between \( (x, y) \) and \( (0, 3) \) is given by the equation: \[ \sqrt{x^2 + (y - 3)^2} = r + 2 \]
Step 5: Use the condition that the circle touches the x-axis.
Since the circle touches the x-axis, the distance from its center to the x-axis is equal to its radius \( r \). Therefore, the equation becomes: \[ y = r \]
Step 6: Conclusion.
By substituting \( y = r \) into the distance equation, we obtain the equation of a parabola. Therefore, the locus of the center of the circle is a parabola, corresponding to option (A).
Step 7: Verification.
By analyzing the geometry and applying the distance conditions, we confirm that the locus is indeed a parabola.
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