find the equation of the perpendicular bisector of a chord of a cicle, given that the end points of the chordare(-2,0) and D(4,-4).
The formulas required are:
1. Mid-point between two given points (x1,y1), (x2,y2) is ((x1+x2)/2,(y1+y2)/2).
2. Slope between two given points:
(y2-y1)/(x2-x1)
3. two lines with slopes m1 and m2 are perpendicular if m1*m2=-1.
Can you work out the answer with the above information?
first, get the equation of the chord:
slope, m = (y2-y1)/(x2-x1)
m = (0+4)/(-2-4) = -2/3
slope of the line: y - y1 = m(x - x1)
*just choose one point (for y1 and x1 to substitute),, in this case, i chose the first point:
y - 0 = (-2/3)(x + 2)
*transform this into standard form (Ax + By = C):
3y = -2(x+2)
3y = -2x - 4
2x + 3y = -4
then get the midpoint of the 2 given points (because perpendicular bisector passes through the midpoint):
midpoint = ((x1+x2)/2 , (y1+y2)/2)
midpoint = ((-2+4)/2 , (0-4)/2)
midpoint = (1,-2)
one shortcut to get the equation of perpendicular bisector: using the equation of chord that we previously calculated, exchange the positions of A and B and reverse the sign:
2x + 3y = -4 becomes
3x - 2y = C
*we calculate C by substituting the midpoint to the equation, thus:
3(1) - 2(-2) = C
3 + 4 = 7 = C
therefore:
3x - 2y = 7
*another solution would be after getting the slope, get its negative reciprocal,, then get the equation of the line using the new slope, and the midpoint,, you should get the same result~
so there,, :)
To find the equation of the perpendicular bisector of a chord of a circle, we need to use the midpoint formula and the slope formula.
1. Start by finding the midpoint of the chord:
- Given the endpoints of the chord, (-2, 0) and (4, -4), we can find the midpoint using the midpoint formula:
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Midpoint = ((-2 + 4)/2, (0 + (-4))/2)
Midpoint = (1, -2)
2. Next, calculate the slope of the chord:
- Given the endpoints of the chord, we can find the slope using the slope formula:
Slope = (y₂ - y₁)/(x₂ - x₁)
Slope = (-4 - 0)/(4 - (-2))
Slope = -4/6
Simplifying, we get Slope = -2/3
3. Since the perpendicular bisector is perpendicular to the chord, its slope will be the negative reciprocal of the chord's slope. In this case, the perpendicular bisector will have a slope of 3/2.
4. Now that we have the slope and the midpoint, we can use the point-slope form of a line to find the equation of the perpendicular bisector:
- Using the point-slope form: y - y₁ = m(x - x₁)
- Plugging in the values: y - (-2) = (3/2)(x - 1)
- Simplifying, we get: y + 2 = (3/2)(x - 1)
- Distributing, we get: y + 2 = (3/2)x - 3/2
- Rearranging, we get: y = (3/2)x - 3/2 - 2
- Simplifying further, we get y = (3/2)x - 7/2
So, the equation of the perpendicular bisector of the chord with endpoints (-2, 0) and (4, -4) is y = (3/2)x - 7/2.