- The equation of the line through A and B is equal to y = -4x/3 + 2.
- The perpendicular bisector of the line AB is equal to 3x - 4y = 17 (Proved).
- The equation of this circle is equal to (x - 15)² + (y - 7)² = 325
Given the following data:
Points on x-axis = (-3, 9).
Points on y-axis = (6, -10).
What is a slope?
The slope of a line simply refers to the gradient of a line and it's typically used to describe both the ratio, direction and steepness of an equation of a straight line.
How to calculate the slope of a line?
Mathematically, the slope of a straight line can be calculated by using this formula;
[tex]Slope, m = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope, m = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]
Substituting the given parameters into the formula, we have;
Slope, m = (-10 - 6)/(9 - (-3))
Slope, m = -16/12
Slope, m = -4/3.
Mathematically, the standard form of the equation of a straight line is given by;
y - y₁ = m(x - x₁)
y - 6 = -4/3(x - (-3))
3y - 18 = -4x - 12
3y = -4x - 12 + 18
3y = -4x + 6
y = -4x/3 + 2.
How to show that the perpendicular bisector of the line AB is 3x-4y=17?
First of all, we would determine the midpoint of line AB as follows:
Midpoint on x-coordinate is given by:
Midpoint = (x₁ + x₂)/2
Midpoint = (-3 + 9)/2
Midpoint = 6/2
Midpoint = 3.
Midpoint on y-coordinate is given by:
Midpoint = (y₁ + y₂)/2
Midpoint = (6 - 10)/2
Midpoint = -4/2
Midpoint = -2.
For perpendicularity, we have:
m₁ × m₂ = -1
-4/3 × m₂ = -1
m₂ = 3/4.
Next, we would use the point-slope form to write the equation:
y - y₁ = m₂(x - x₁)
y - (-2) = 3/4(x - 3)
y + 2 = 3/4(x - 3)
4y + 8 = 3x - 9
3x - 4y = 8 + 9
3x - 4y = 17 (Proved).
The equation of a circle.
Mathematically, the standard form of the equation of a circle is given by;
(x - h)² + (y - k)² = r²
(x - 15)² + (y - 7)² = 325
Read more on point-slope form here: brainly.com/question/24907633
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Complete Question:
A is the point (-3, 6) and B is the point (9, -10).
a) Find the equation of the line through A and B.
b) Show that the perpendicular bisector of the line AB is 3x-4y=17.
c) A circle passes through A and B and has its center on line x=15. Find the equation of this circle.
