To determine if the point [tex]\((3, -1)\)[/tex] lies on the circle given by the equation [tex]\( (x+1)^2 + (y-1)^2 = 16 \)[/tex], we substitute the coordinates of the point into the equation and check if the resulting statement is true. Here's the step-by-step process:
1. Circle's Equation:
[tex]\[(x+1)^2 + (y-1)^2 = 16\][/tex]
2. Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -1\)[/tex] into the equation:
[tex]\[
(3 + 1)^2 + (-1 - 1)^2
\][/tex]
3. Simplify the expression inside the parentheses first:
[tex]\[
(4)^2 + (-2)^2
\][/tex]
4. Compute the squares:
[tex]\[
16 + 4
\][/tex]
5. Add the results:
[tex]\[
20
\][/tex]
6. Compare the result with the right-hand side of the equation [tex]\( (x+1)^2 + (y-1)^2 = 16 \)[/tex]:
[tex]\[
20 \neq 16
\][/tex]
Since the left-hand side of the equation [tex]\( 20 \)[/tex] does not equal the right-hand side [tex]\( 16 \)[/tex], the point [tex]\((3, -1)\)[/tex] does not satisfy the circle's equation.
Conclusion: The point [tex]\((3, -1)\)[/tex] does not lie on the circle. Therefore, the correct option is:
No; when you plug in the point for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the equation, you do not get a true statement.
