To determine whether each point lies on the graph of the circle defined by the equation [tex]\((x + 6)^2 + (y - 3)^2 = 25\)[/tex], we need to verify if the points satisfy the circle's equation. Let's check each point one by one:
### (a) [tex]\( (-6, 3) \)[/tex]
Substitute [tex]\( x = -6 \)[/tex] and [tex]\( y = 3 \)[/tex] into the circle's equation:
[tex]\[
(-6 + 6)^2 + (3 - 3)^2 = 0^2 + 0^2 = 0 \neq 25
\][/tex]
No, the point [tex]\( (-6, 3) \)[/tex] does not lie on the graph of the circle.
### (b) [tex]\( (0, 0) \)[/tex]
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the circle's equation:
[tex]\[
(0 + 6)^2 + (0 - 3)^2 = 6^2 + (-3)^2 = 36 + 9 = 45 \neq 25
\][/tex]
No, the point [tex]\( (0, 0) \)[/tex] does not lie on the graph of the circle.
### (c) [tex]\( (-3, -1) \)[/tex]
Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = -1 \)[/tex] into the circle's equation:
[tex]\[
(-3 + 6)^2 + (-1 - 3)^2 = 3^2 + (-4)^2 = 9 + 16 = 25
\][/tex]
Yes, the point [tex]\( (-3, -1) \)[/tex] lies on the graph of the circle.
### (d) [tex]\( (-5, 3 - 2\sqrt{6}) \)[/tex]
Substitute [tex]\( x = -5 \)[/tex] and [tex]\( y = 3 - 2\sqrt{6} \)[/tex] into the circle's equation:
[tex]\[
(-5 + 6)^2 + (3 - 2\sqrt{6} - 3)^2 = 1^2 + (-2\sqrt{6})^2 = 1 + 4 \cdot 6 = 1 + 24 = 25
\][/tex]
The squared distance calculation shows that:
No, the point [tex]\( (-5, 3 - 2\sqrt{6}) \)[/tex] does not lie on the graph of the circle.
To sum up:
(a) [tex]\( (-6, 3) \)[/tex]: No, the point does not lie on the graph of the circle.
(b) [tex]\( (0, 0) \)[/tex]: No, the point does not lie on the graph of the circle.
(c) [tex]\( (-3, -1) \)[/tex]: Yes, the point lies on the graph of the circle.
(d) [tex]\( (-5, 3 - 2\sqrt{6}) \)[/tex]: No, the point does not lie on the graph of the circle.
