Find all values of a such that y=a/(x−9) and y=x^2−18x+81 intersect at right angles?
please how the steps :)
y = a/(x-9) = a(x-9)^-1
dy/dx -a(x-9)^-2 = -a/(x-9)^2
y = x^2 - 18x + 81 = (x-9)^2
dy/dx = 2(x - 9)
recall that for perpendicular lines their slopes are negative reciprocals of each other
(x-9)^2/a = 2(x-9) or a = (x-9)^2 / (2(x-9))
a = (x-9)/2
Where do they intersect ?
a/(x-9) = (x-9)^2
a =(x-9)^3
then
(x-9)^3 = (x-9)/2
(x-9)^2 = 1/2
x-9 = ± 1/√2 or ± √2/2
sub into a = (x-9)/2 = ± √2/4
I don't think the above answer is valid because you will continue getting the same answer of ± √2/4 no matter the question.
for example y=a/(x−8) and y=x^2−16x+64, doing it with the same method will receive the same value. I've tried 3 times and continue getting ± √2/4. Unless I've done one of your steps wrong, I don't think this method is valid.
To find the values of 'a' for which the curves represented by the equations y = a/(x-9) and y = x^2 - 18x + 81 intersect at right angles, we need to determine the slope of each curve at their point of intersection.
Step 1: Find the slope of the first curve, y = a/(x-9):
To find the slope of a curve given by a function f(x), we can differentiate it with respect to x.
Differentiating y = a/(x-9), we get:
dy/dx = -a/(x-9)^2
Step 2: Find the slope of the second curve, y = x^2 - 18x + 81:
To find the slope of a curve given by a quadratic equation ax^2 + bx + c, we can differentiate it with respect to x.
Differentiating y = x^2 - 18x + 81, we get:
dy/dx = 2x - 18
Step 3: Set the slopes of the curves equal to each other:
Since the curves intersect at right angles, their slopes will be negative reciprocals of each other.
So, -a/(x-9)^2 = -1 / (2x - 18)
Step 4: Solve the equation for 'x':
Multiply both sides by (2x - 18)(x-9)^2 to eliminate the denominators:
- a(2x - 18) = -1(x-9)^2
-2ax + 18a = -(x-9)(x-9)
-2ax + 18a = -(x^2 - 18x + 81)
-2ax + 18a = -x^2 + 18x - 81
Rearrange to form a quadratic equation:
x^2 + (-2a+18)x + (18a-81) = 0
Step 5: Determine the discriminant of the quadratic equation:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula b^2 - 4ac.
For our equation, the discriminant is:
(-2a + 18)^2 - 4(18a - 81)
Step 6: Set the discriminant equal to zero:
To find the values of 'a' for which the curves intersect at right angles, the discriminant should be zero. So, we solve:
(-2a + 18)^2 - 4(18a - 81) = 0
Step 7: Solve for 'a':
Expand and simplify the equation:
4a^2 - 72a + 324 - 72a + 324 = 0
4a^2 - 144a + 648 = 0
Factorize the quadratic equation:
(a - 9)(4a - 72) = 0
Solve for 'a':
Setting each factor equal to zero:
a - 9 = 0 --> a = 9
4a - 72 = 0 --> a = 18
Therefore, the values of 'a' for which the curves intersect at right angles are a = 9 and a = 18.