Let a and b be real numbers, where a<b, and let A= (a,a^2) and B=(b,b^2). The line AB (meaning the unique line that contains the point A and the point B) has x-intercept (-3/2,0) and y-intercept (0,3). Find a and b.
Are we to assume that your line AB is a straight line?
If so, we can find the slope of AB
slope AB = (b^2 -a^2)/(b-a)
= (b-a)(b+a)/(b-a)
= b+a
we can write the equation as
y - a^2 = (b+a)(x-a)
also (3/2,0) lies on it, so
0-a^2 = (b+a)(-3/2 - a)
-a^2 = -3b/2 -ab - 3a/2 - a^2
3b/2 - 3a/2 = -ab
3a + 3b = -2ab , #1
and (0,3) lies on it
3 - a^2 = (b+a)(-a)
3-a^2 = -ab - a^2
then ab = -3 or a = -3/b
sub that into #1
3(-3/b) + 3b = -2(-3)
-3/b + b = 2
times b
-3 + b^2 = 2b
b^2 - 2b - 3 = 0
(b-3)(b+1) = 0
b = 3 or b = -1
if b = 3 , a = -3/3 = -1
if b = -1 , a = -3/-1 = 3 , but a should be < b
so a = -1 and b = 3
-1,3
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To find the values of a and b, we can make use of the information about the x-intercept and y-intercept of the line AB.
The x-intercept is given as (-3/2, 0), which means that the line AB intersects the x-axis at x = -3/2. Since the x-coordinate is -3/2, we can set up the equation for the line AB:
y = mx + c, where m is the slope of the line and c is the y-intercept.
Substituting the x-coordinate (-3/2) and y-coordinate (0) of the x-intercept, we have:
0 = m(-3/2) + c
Simplifying this equation gives us:
0 = -3m/2 + c
c = 3m/2
Next, let's consider the y-intercept, which is given as (0, 3). This means that the line AB intersects the y-axis at y = 3. Substituting these values into the equation for the line AB, we have:
3 = m(0) + c
3 = c
Now that we have found the value of the y-intercept c, we can substitute it back into the equation we obtained from the x-intercept:
0 = -3m/2 + c
0 = -3m/2 + 3
To simplify further, let's multiply the equation by 2 to get rid of the fraction:
0 = -3m + 6
Rearranging the equation gives us:
3m = 6
m = 2
We now have the slope of the line, which is 2. Substituting this into the equation for the line AB, we have:
y = 2x + 3
Now we can find the x-coordinate of the point B, which lies on the line AB:
b = -3/2
To find the corresponding y-coordinate, we substitute b into the equation we obtained:
b^2 = 2b + 3
Simplifying further, we have:
b^2 - 2b - 3 = 0
Factoring the equation or using the quadratic formula will give us the two possible solutions for b. From there, we can substitute each value back into the equation for the line AB to find the corresponding values of a.