Which point is on the circle centered at the origin with a radius of 5 units?
Distance formula: [tex]$\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$[/tex]
A. [tex]$(2, \sqrt{21})$[/tex]
B. [tex]$(2, \sqrt{23})$[/tex]
C. [tex]$(2,1)$[/tex]
D. [tex]$(2,3)$[/tex]
Answer (2)
The equation of a circle centered at the origin with radius 5 is x 2 + y 2 = 25 .
Check if the point ( 2 , 21 ) lies on the circle: 2 2 + ( 21 ) 2 = 4 + 21 = 25 .
Check if the point ( 2 , 23 ) lies on the circle: 2 2 + ( 23 ) 2 = 4 + 23 = 27 .
The point that lies on the circle is ( 2 , 21 ) .
Explanation
Problem Analysis We are given a circle centered at the origin (0,0) with a radius of 5. The equation of this circle is x 2 + y 2 = r 2 , where r is the radius. Since the radius is 5, the equation becomes x 2 + y 2 = 5 2 = 25 . We need to determine which of the given points satisfies this equation.
Checking point (2, sqrt(21)) Let's check each point to see if it lies on the circle:
Point ( 2 , 21 ) :
We need to calculate 2 2 + ( 21 ) 2 .
Result for (2, sqrt(21)) 2 2 + ( 21 ) 2 = 4 + 21 = 25 . Since this equals 25, the point ( 2 , 21 ) lies on the circle.
Checking point (2, sqrt(23))
Point ( 2 , 23 ) :
We need to calculate 2 2 + ( 23 ) 2 .
Result for (2, sqrt(23)) 2 2 + ( 23 ) 2 = 4 + 23 = 27 . Since this does not equal 25, the point ( 2 , 23 ) does not lie on the circle.
Checking point (2, 1)
Point ( 2 , 1 ) :
We need to calculate 2 2 + 1 2 .
Result for (2, 1) 2 2 + 1 2 = 4 + 1 = 5 . Since this does not equal 25, the point ( 2 , 1 ) does not lie on the circle.
Checking point (2, 3)
Point ( 2 , 3 ) :
We need to calculate 2 2 + 3 2 .
Result for (2, 3) 2 2 + 3 2 = 4 + 9 = 13 . Since this does not equal 25, the point ( 2 , 3 ) does not lie on the circle.
Final Answer Therefore, the only point that lies on the circle is ( 2 , 21 ) .
Examples
Understanding circles and their equations is crucial in many real-world applications. For instance, when designing a circular garden, you need to ensure that all points along the edge are equidistant from the center to maintain a perfect circular shape. Similarly, in architecture, circular arches and domes rely on the principles of circular geometry to distribute weight evenly and maintain structural integrity. In navigation, the concept of a circle is used to define the range of a radar system or the coverage area of a radio tower, ensuring that all points within a certain radius receive the signal.
The point that lies on the circle centered at the origin with a radius of 5 units is ( 2 , 21 ) . This was determined by checking each point against the circle's equation, x 2 + y 2 = 25 . Only the point ( 2 , 21 ) satisfied this equation.
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