Find the equation of the chord of the parabola joining the points with parameters
(a)1 and -3 on x=2t, y=t^2
(b)1/2 and 2 on x=4t, y=2t^2
(c)-1 and -2 on x=t, y=1/2t^2
(d)-2 and 4 on x=1/2t^2, y=1/4t^2
Then use the formula y=1/2(p+q)x-apq to obtain the chords in parts (a)-(d)
PLEASE&THANKY0U VERY MUCH :)
(d) is not a parabola since both x and y are proportional to the parameter t^2. Did you copy the problem correctly?
These are four separate problems that can be all done the same way. Consider (a):
t = x/2, so y = (1/4) x^2 is the equation of the parabola.
When t = 1, x = 2 and y = 1
When t = -3, x = -6 and y = 9
The chord that they want connects these two points.
The slope is 8/(-8) = -1
y = -x + b
1 = -2 + b
b = 3
y = -x + 3 is the chord equation, with
-6 < x < 2
You need to define a, p and q in the equation
y=(1/2)(p+q)x-apq
That is not the equation of a parabola.
To find the equation of the chord of a parabola, we need to find the coordinates of the two points that define the chord. Let's solve each part one by one:
(a) Given x = 2t and y = t^2 and two points with parameters 1 and -3, we can substitute these values into the equation of the parabola to find the corresponding coordinates:
For t = 1, we have x = 2(1) = 2 and y = (1)^2 = 1. So the first point is (2, 1).
For t = -3, we have x = 2(-3) = -6 and y = (-3)^2 = 9. So the second point is (-6, 9).
Now we can use these two points to find the equation of the chord. Using the point-slope form, we have:
(y - y1) = m(x - x1), where (x1, y1) are the coordinates of the first point.
The slope, m, is given by (y2 - y1) / (x2 - x1), where (x2, y2) are the coordinates of the second point. Substituting the values, we get:
m = (9 - 1) / (-6 - 2) = 8 / -8 = -1
Using the first point (2, 1) and the slope -1, we can now write the equation of the chord:
(y - 1) = -1(x - 2)
y - 1 = -x + 2
y = -x + 3
So the equation of the chord is y = -x + 3.
We can repeat the same process for parts (b), (c), and (d) to find the equations of the chords:
(b) Given x = 4t and y = 2t^2, and the points with parameters 1/2 and 2, we find the first point as (2, 1/2) and the second point as (8, 8).
The slope, m, is (8 - 1/2) / (8 - 2) = 15/6 = 5/2.
Using the first point (2, 1/2) and the slope 5/2, the equation of the chord is y = (5/2)x - 4.
(c) Given x = t and y = (1/2)t^2, and the points with parameters -1 and -2, we find the first point as (-1, 1/2) and the second point as (-2, 2).
The slope, m, is (2 - 1/2) / (-2 + 1) = 3/2.
Using the first point (-1, 1/2) and the slope 3/2, the equation of the chord is y = (3/2)x + 2.
(d) Given x = 1/2t^2 and y = 1/4t^2, and the points with parameters -2 and 4, we find the first point as (1/2, 1/4) and the second point as (8, 1/4).
The slope, m, is (1/4 - 1/4) / (8 - 1/2) = 0.
Using the first point (1/2, 1/4) and the slope 0, the equation of the chord is y = 1/4.
Using the formula y = (1/2)(p + q)x - apq, we can cross-verify the results:
(a) Given p = 1, q = -3, and a = 2, the equation y = (1/2)(1 - 3)x - (2)(1)(-3) becomes y = -x + 3, which matches our previous result.
(b) Given p = 1/2, q = 2, and a = 4, the equation y = (1/2)(1/2 + 2)x - (4)(1/2)(2) becomes y = (5/2)x - 4, which matches our previous result.
(c) Given p = -1, q = -2, and a = 1, the equation y = (1/2)(-1 - 2)x - (1)(-1)(-2) becomes y = (3/2)x + 2, which matches our previous result.
(d) Given p = -2, q = 4, and a = 1/2, the equation y = (1/2)(-2 + 4)x - (1/2)(-2)(4) becomes y = (1/4)x, which matches our previous result.
Thus, the equations obtained using the formula also match the equations found earlier.