a) The coordinates of the point Q is Q(x, y) = (- 2, - 3).
b) The equation of the bisector is y = - (1 / 2) · x + 19 / 4.
c) The points of intersection of the bisector with the quadratic equation are (0.991, 4.255) and (- 5.741, 7.621).
<h3>How to analyze a system formed by a quadratic equation and a linear equation</h3>
In this problem we find a system formed by a linear equation and a quadratic equation, in which they intersect each other twice. a) According to the statement, the line intersects the parabola at the point P(x, y) = (5, 11), then the slope of the equation of the line is:
11 = 5 · m + 1
5 · m = 10
m = 2
Then, the equation of the line is y = 2 · x + 1 and we eliminate y by both equations:
y² = 51 + 19 · x - x²
(2 · x + 1)² = 51 + 19 · x - x²
4 · x² + 4 · x + 1 = 51 + 19 · x - x²
5 · x² - 15 · x - 50 = 0
5 · (x² - 3 · x - 10) = 0
5 · (x - 5) · (x + 2) = 0
The x-coordinate of the second point of intersection is - 2 and the y-coordinate is:
y = 2 · (- 2) + 1
y = - 3
The coordinates of the point Q is Q(x, y) = (- 2, - 3).
b) A bisector is a perpendicular line that partitions a line segment into two segments of equal length. First, we find the coordinates of the midpoint of the segment PQ:
M(x, y) = 0.5 · P(x, y) + 0.5 · Q(x, y)
M(x, y) = 0.5 · (5, 11) + 0.5 · (- 2, - 3)
M(x, y) = (1.5, 4)
Second, we calculate the slope of the bisector:
m = - 1 / 2
Third, we find the intercept of the bisector:
4 = - (1 / 2) · (1.5) + b
b = 4.75
Then, the equation of the bisector is y = - (1 / 2) · x + 19 / 4.
c) Then, we eliminate the variable y in both equations:
[- (1 / 2) · x + 19 / 4]² = 51 + 19 · x - x²
(1 / 4) · x² - (19 / 4) · x + 361 / 16 = 51 + 19 · x - x²
(5 / 4) · x² - (95 / 4) · x - (455 / 16) = 0
80 · x² + 380 · x - 455 = 0
(16 · 5) · x² + (4 · 5 · 19) · x - (5 · 7 · 13) = 0
16 · x² + 76 · x - 91 = 0
(4 · x)² + 19 · (4 · x) - 91 = 0
(4 · x - 3.962) · (4 · x + 22.963) = 0
There are two x-coordinates: x₁ = 0.991, x₂ = - 5.741. And the y-coordinates are:
y₁ = - (1 / 2) · (0.991) + 19 / 4
y₁ = 4.255
y₂ = - (1 / 2) · (- 5.741) + 19 / 4
y₂ = 7.621
The points of intersection of the bisector with the quadratic equation are (0.991, 4.255) and (- 5.741, 7.621).
To learn more on equations of the line: brainly.com/question/2564656
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