Find all values of a such that y=a/(x−8) and y=x^2−16x+64 intersect at right angles?
please how the steps :)
The curves intersect where
a/(x-8) = (x-8)^2
a = (x-8)^3
x = 8 + cbrt(a)
Now, we need to find a such that the curves are perpendicular.
slope of x^2 - 16x + 64 = 2x - 16
slope of a/(x-8) = -a/(x-8)^2
perpendicular slope = (x-8)^2/a
So, when x = 8 + cbrt(a), and the slopes are equal, we have
2(8+cbrt(a) - 16 = cbrt(a)^2 / a
16 - 2cbrt(a) - 16 = 1/cbrt(a)
2cbrt(a) = 1/cbrt(a)
a = 2^(-3/2)
cbrt(a) = 1/√2
Checking the slopes at x = 8 + 1/√2
2x - 16 = 2(8+1/√2) - 16 = 16 - √2 - 16 = √2
(x-8)^2/a = 1/2 / 2^(-3/2) = 2^(3/2)/2 = √2
I don't get how you solved 2^(-3/2)and got 1/sqrt2
thanks steve :)
2(8+cbrt(a) - 16 = cbrt(a)^2 / a
where did cbrt(a)^2 / a come from??
and how did you go from cbrt(a)^2 / a into 1/cbrt(a) ????
To find the values of a such that the two curves intersect at right angles, we need to determine where the derivatives of the curves are perpendicular to each other.
Step 1: Determine the derivative of each equation:
- For the equation y = a / (x - 8), we will use the quotient rule to find its derivative. Let's call it f(x):
f(x) = a / (x - 8)
f'(x) = (a * (x - 8)' - (a * (x - 8))') / ((x - 8)^2)
= a / (x - 8)^2
- For the equation y = x^2 − 16x + 64, let's call it g(x). We can differentiate it to find its derivative:
g(x) = x^2 − 16x + 64
g'(x) = 2x - 16
Step 2: Determine where the derivatives are perpendicular:
Since two curves are perpendicular when the slopes of their tangent lines are negative reciprocals, we can equate the two derivatives and find the values for x when this condition is met.
a / (x - 8)^2 = -1 / (2x - 16)
To simplify the equation, multiply both sides by (2x - 16) and (x - 8)^2:
a * (2x - 16) = -1 * (x - 8)^2
Expand and simplify:
2ax - 16a = -(x^2 - 16x + 64)
2ax - 16a = -x^2 + 16x - 64
Rearrange the equation and set it equal to zero:
x^2 - 2ax + 16x + 16a - 64 = 0
x^2 + (16 - 2a)x + (16a - 64) = 0
Step 3: Determine the discriminant:
The discriminant (b^2 - 4ac) will help us determine the nature of the roots. For perpendicular lines, the discriminant should be zero.
The discriminant formula for the quadratic equation ax^2 + bx + c = 0 is:
D = b^2 - 4ac
For our quadratic equation, the discriminant becomes:
D = (16 - 2a)^2 - 4(16a - 64)
Step 4: Solve for a:
Set the discriminant equal to zero and solve for a:
(16 - 2a)^2 - 4(16a - 64) = 0
Expand and simplify:
256 + 4a^2 - 64a - 4(16a - 64) = 0
256 + 4a^2 - 64a - 64a + 256 = 0
4a^2 - 128a + 512 = 0
Divide both sides by 4 to simplify the equation:
a^2 - 32a + 128 = 0
Step 5: Solve for a using the quadratic formula or factoring:
Using the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / 2a
Apply the values from our equation:
a = (-(-32) ± √((-32)^2 - 4(1)(128))) / 2(1)
a = (32 ± √(1024 - 512)) / 2
a = (32 ± √512) / 2
a = (32 ± 16√2) / 2
a = 16 ± 8√2
Therefore, the values of a that make the two curves intersect at right angles are a = 16 + 8√2 and a = 16 - 8√2.