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A circle touches both the yaxis and the line x y=0.Then locus of its center?
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A circle touches both the yaxis and the line x y=0.Then locus of its c...
The Locus of the Center of the Circle
Let's consider a circle that touches both the y-axis and the line xy = 0. We need to determine the locus of its center, which means finding the set of all possible points that the center of the circle can be.
Understanding the Problem
Before we dive into solving the problem, let's understand what it means for a circle to touch the y-axis and the line xy = 0.
If a circle touches the y-axis, it means that the distance between the center of the circle and the y-axis is equal to the radius of the circle.
If a circle touches the line xy = 0, it means that the distance between the center of the circle and the line xy = 0 is equal to the radius of the circle.
Approach to Solving
To find the locus of the center of the circle, we need to find the equation that represents all possible points that satisfy the given conditions.
Let's assume the center of the circle is (h, k) and the radius is r.
Since the circle touches the y-axis, the distance between the center (h, k) and the y-axis is equal to the radius r. This can be expressed as:
|h| = r (Equation 1)
Since the circle touches the line xy = 0, the distance between the center (h, k) and the line xy = 0 is equal to the radius r. This can be expressed as:
|k| = r (Equation 2)
Solving the Equations
We have two equations: Equation 1 and Equation 2. Let's solve them separately.
Solving Equation 1: |h| = r
If h is positive, then |h| = h. In this case, Equation 1 becomes:
h = r (Equation 3)
If h is negative, then |h| = -h. In this case, Equation 1 becomes:
-h = r (Equation 4)
Combining both cases, we can write Equation 1 as:
|h| = r (Equation 5)
Solving Equation 2: |k| = r
If k is positive, then |k| = k. In this case, Equation 2 becomes:
k = r (Equation 6)
If k is negative, then |k| = -k. In this case, Equation 2 becomes:
-k = r (Equation 7)
Combining both cases, we can write Equation 2 as:
|k| = r (Equation 8)
The Locus of the Center of the Circle
The locus of the center of the circle is the set of all possible points (h, k) that satisfy both Equation 5 and Equation 8.
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A circle touches both the yaxis and the line x y=0.Then locus of its center?
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A circle touches both the yaxis and the line x y=0.Then locus of its center? 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 both the yaxis and the line x y=0.Then locus of its center? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle touches both the yaxis and the line x y=0.Then locus of its center?.
A circle touches both the yaxis and the line x y=0.Then locus of its center? 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 both the yaxis and the line x y=0.Then locus of its center? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle touches both the yaxis and the line x y=0.Then locus of its center?.
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