The equation x² + y² = 25 defines a circle with center at the origin and radius 5. The line y = x - 1 passes
through the circle. Using the substitution method, find the point(s) at which the circle and the line intersect.
a. (4, 3) and (–4, –3) c. (4, –3) and (–3, 4)
b. (3, 4) and (–3, –4) d. no solution
Substitute y = x - 1 into x² + y² = 25:
x² + (x - 1)² = 25
Simplifying:
2x² - 2x - 24 = 0
x² - x - 12 = 0
Factoring:
(x - 4)(x + 3) = 0
So x = 4 or x = -3.
Substituting each value into y = x - 1 gives y = 3 for x = 4, and y = -4 for x = -3.
Therefore, the points of intersection are (4, 3) and (-3, -4), which corresponds to option b.
To find the intersection point(s) of the circle and the line, we can substitute the equation of the line (y = x - 1) into the equation of the circle (x² + y² = 25).
Substituting y = x - 1 into x² + y² = 25:
x² + (x - 1)² = 25
Expanding the equation:
x² + x² - 2x + 1 = 25
Combining like terms:
2x² - 2x + 1 - 25 = 0
Simplifying:
2x² - 2x - 24 = 0
Dividing through by 2:
x² - x - 12 = 0
Now we can factorize or use the quadratic formula to find the values of x:
(x - 4)(x + 3) = 0
Setting each factor equal to zero:
x - 4 = 0 or x + 3 = 0
x = 4 or x = -3
Now we substitute these x values back into the line equation to find the corresponding y values:
For x = 4:
y = 4 - 1
y = 3
For x = -3:
y = -3 - 1
y = -4
Therefore, the intersection points of the circle and the line are (4, 3) and (-3, -4).
So, the correct answer is b. (3, 4) and (–3, –4).