The equation of a circle with center (5, 4) and touches the y-axis is?
Understand the Problem
The question is asking for the equation of a circle given its center and a condition regarding its position relative to the y-axis. The center is specified as (5, 4) and the circle is said to touch the y-axis.
Answer
$$(x - 5)^2 + (y - 4)^2 = 25$$
Answer for screen readers
The equation of the circle is: $$(x - 5)^2 + (y - 4)^2 = 25$$
Steps to Solve
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Identify the center and radius The center of the circle is given as ( (5, 4) ). Since the circle touches the y-axis, the distance from the center to the y-axis is equal to the radius ( r ). The distance from the center to the y-axis is the x-coordinate, which is 5. Therefore, the radius ( r = 5 ).
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Write the standard equation of the circle The standard form of the equation of a circle with center ( (h, k) ) and radius ( r ) is given by: $$ (x - h)^2 + (y - k)^2 = r^2 $$ In this case, substituting ( h = 5 ), ( k = 4 ), and ( r = 5 ): $$ (x - 5)^2 + (y - 4)^2 = 5^2 $$
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Simplify the equation We can simplify the equation to make it more concise. Calculate ( 5^2 ): $$ 5^2 = 25 $$ So the equation becomes: $$ (x - 5)^2 + (y - 4)^2 = 25 $$
The equation of the circle is: $$(x - 5)^2 + (y - 4)^2 = 25$$
More Information
This equation represents a circle with a center at ( (5, 4) ) and a radius of 5. Since the circle touches the y-axis, it confirms that the distance from the center to the y-axis is equal to the radius.
Tips
- Misunderstanding the radius: One common mistake might be assuming the radius is equal to the y-coordinate of the center instead of the x-coordinate when the circle touches the y-axis. To avoid this, remember that the radius in this scenario is the distance from the center to the y-axis, which is the x-coordinate.
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