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The equation of the circle, passing through the origin such that the x-axis is its diameter, is given by
- a)x2 + y2 - 2ky = 0
- b)x2 + y2 - 2hx = 0
- c)x2 + y2 - 2hx + h2 = 0
- d)x2 + y2 - 2ky + k2 = 0
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The equation of the circle, passing through the origin such that the x...
Since the circle passes through the origin and has x-axis as its diameter, therefore its centre is [h, 0) and radius = h
∴ Required equation of the circle is
(x-h)2 + (y-0)2 = h2
or x2 + y2 -2hx = 0
∴ Required equation of the circle is
(x-h)2 + (y-0)2 = h2
or x2 + y2 -2hx = 0
Most Upvoted Answer
The equation of the circle, passing through the origin such that the x...
Understanding the Circle's Properties
To find the equation of a circle that passes through the origin and has the x-axis as its diameter, we need to analyze the properties of the circle.
Circle with x-axis as Diameter
- A circle with the x-axis as its diameter implies that its center lies on the x-axis.
- The radius of the circle extends equally in both the positive and negative y-directions.
General Equation of the Circle
The general equation of a circle with center at (h, k) and radius r is:
- (x - h)² + (y - k)² = r²
Given that the x-axis is the diameter, the center will be (h, 0), and the radius will be the distance from the center to the x-axis, which is |k|.
Condition of Passing Through the Origin
Since the circle passes through the origin (0,0), substituting these coordinates into the circle equation gives:
- (0 - h)² + (0 - 0)² = r²
- This simplifies to h² = r².
Substituting for Radius
With the radius being the distance from the center to the x-axis (|k|), we can express the equation as:
- h² = k²
Thus, the equation of the circle can be reformulated as:
- x² + y² - 2hx = 0
This matches option 'B'.
Conclusion
The correct equation of the circle that passes through the origin and has the x-axis as its diameter is indeed given by option 'B':
- x² + y² - 2hx = 0.
To find the equation of a circle that passes through the origin and has the x-axis as its diameter, we need to analyze the properties of the circle.
Circle with x-axis as Diameter
- A circle with the x-axis as its diameter implies that its center lies on the x-axis.
- The radius of the circle extends equally in both the positive and negative y-directions.
General Equation of the Circle
The general equation of a circle with center at (h, k) and radius r is:
- (x - h)² + (y - k)² = r²
Given that the x-axis is the diameter, the center will be (h, 0), and the radius will be the distance from the center to the x-axis, which is |k|.
Condition of Passing Through the Origin
Since the circle passes through the origin (0,0), substituting these coordinates into the circle equation gives:
- (0 - h)² + (0 - 0)² = r²
- This simplifies to h² = r².
Substituting for Radius
With the radius being the distance from the center to the x-axis (|k|), we can express the equation as:
- h² = k²
Thus, the equation of the circle can be reformulated as:
- x² + y² - 2hx = 0
This matches option 'B'.
Conclusion
The correct equation of the circle that passes through the origin and has the x-axis as its diameter is indeed given by option 'B':
- x² + y² - 2hx = 0.
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Community Answer
The equation of the circle, passing through the origin such that the x...
Since the circle passes through the origin and has x-axis as its diameter, therefore its centre is [h, 0) and radius = h
∴ Required equation of the circle is
(x-h)2 + (y-0)2 = h2
or x2 + y2 -2hx = 0
∴ Required equation of the circle is
(x-h)2 + (y-0)2 = h2
or x2 + y2 -2hx = 0
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The equation of the circle, passing through the origin such that the x-axis is its diameter, is given bya)x2 + y2 - 2ky = 0b)x2 + y2 - 2hx= 0c)x2 + y2 - 2hx+ h2 = 0d)x2 + y2 - 2ky + k2 = 0Correct answer is option 'B'. Can you explain this answer?
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