Select the correct answer.
Which point lies on the circle represented by the equation $(x-3)^2+(y+4)^2=6^2$?
A. $(9,-2)$
B. $(0,11)$
C. $(3,10)$
D. $(-9,4)$
E. $(-3,-4)$
Answer (1)
Substitute each point's coordinates into the circle's equation.
Evaluate the equation for each point.
Check if the result equals 6 2 = 36 .
Identify the point that satisfies the equation: ( − 3 , − 4 ) .
Explanation
Analyze the problem The equation of the circle is given by ( x − 3 ) 2 + ( y + 4 ) 2 = 6 2 . We need to find which of the given points satisfies this equation. This means we need to substitute the x and y coordinates of each point into the equation and see if the result equals 6 2 = 36 .
Test each point Let's test each point:
A. ( 9 , − 2 ) : Substitute x = 9 and y = − 2 into the equation: ( 9 − 3 ) 2 + ( − 2 + 4 ) 2 = ( 6 ) 2 + ( 2 ) 2 = 36 + 4 = 40 Since 40 e q 36 , point A does not lie on the circle.
B. ( 0 , 11 ) : Substitute x = 0 and y = 11 into the equation: ( 0 − 3 ) 2 + ( 11 + 4 ) 2 = ( − 3 ) 2 + ( 15 ) 2 = 9 + 225 = 234 Since 234 e q 36 , point B does not lie on the circle.
C. ( 3 , 10 ) : Substitute x = 3 and y = 10 into the equation: ( 3 − 3 ) 2 + ( 10 + 4 ) 2 = ( 0 ) 2 + ( 14 ) 2 = 0 + 196 = 196 Since 196 e q 36 , point C does not lie on the circle.
D. ( − 9 , 4 ) : Substitute x = − 9 and y = 4 into the equation: ( − 9 − 3 ) 2 + ( 4 + 4 ) 2 = ( − 12 ) 2 + ( 8 ) 2 = 144 + 64 = 208 Since 208 e q 36 , point D does not lie on the circle.
E. ( − 3 , − 4 ) : Substitute x = − 3 and y = − 4 into the equation: ( − 3 − 3 ) 2 + ( − 4 + 4 ) 2 = ( − 6 ) 2 + ( 0 ) 2 = 36 + 0 = 36 Since 36 = 36 , point E lies on the circle.
Conclusion The point that satisfies the equation ( x − 3 ) 2 + ( y + 4 ) 2 = 6 2 is ( − 3 , − 4 ) . Therefore, the correct answer is E.
Examples
Understanding circles is crucial in many real-world applications. For instance, consider a radio tower that broadcasts signals in a circular pattern. If you know the equation representing the broadcast range, you can determine whether a specific location (point) is within the signal's reach. This helps in planning communication networks and ensuring reliable coverage.
