find all points on the hyperbola y=1/x, that are closest to the origin on (1/2,2)
Let P(x,y) be the closest point
then if D is the distance, then
D^2 = x^2 + y^2
= x^2 + 1/x^2
2D dD/dx = 2x - 2/x^3
= 0 for a min of D
2x = 2/x^3
x^4 = 1
x = ± 1
if x=1, then y = 1
if x = -1, then y = -1
The two points closest to the origin are (1,1) and (-1,-1)
Where does the (1/2,2) come in?
Was that your answer?
To find the points on the hyperbola y=1/x that are closest to the origin, we need to minimize the distance between the origin (0,0) and a point on the hyperbola.
Let's find the distance between a general point (x, y) on the hyperbola and the origin:
Distance = √((x - 0)^2 + (y - 0)^2)
= √(x^2 + y^2)
We want to minimize this distance, so we need to find the minimum value of the function √(x^2 + y^2).
Now, let's substitute y=1/x into the distance formula:
Distance = √(x^2 + (1/x)^2)
= √(x^2 + 1/x^2)
To find the minimum value of this function, we can take the derivative with respect to x and set it equal to zero:
d/dx (√(x^2 + 1/x^2)) = 0
To simplify the process, let's simplify the expression under the square root:
Let u = x^2 + 1/x^2
Now, we can rewrite the expression for the derivative:
d/dx (√u) = 0
Using the chain rule, we get:
(1/2)u^(-1/2)*du/dx = 0
u^(-1/2)*du/dx = 0
Now, let's find du/dx:
du/dx = d/dx (x^2 + 1/x^2)
= 2x - 2/x^3
Substituting back u = x^2 + 1/x^2, we have:
u^(-1/2)*(2x - 2/x^3) = 0
Since u^(-1/2) is never zero, the equation simplifies to:
2x - 2/x^3 = 0
Now, let's solve this equation for x:
2x = 2/x^3
2x^4 = 2
x^4 = 1
x = ±1
So, the potential x-values for the points closest to the origin lie at x = ±1.
Now, substitute these x-values back into y=1/x to find the corresponding y-values:
For x = 1: y = 1/1 = 1
For x = -1: y = 1/(-1) = -1
Therefore, the two points on the hyperbola y=1/x that are closest to the origin are (1, 1) and (-1, -1).
To find the points on the hyperbola y = 1/x that are closest to the origin, you can use the concept of distance. The distance formula between two points in a coordinate plane is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, we want to minimize the distance between the origin (0, 0) and a point on the hyperbola. Let's denote the x-coordinate of the point on the hyperbola as 'a'. Since y = 1/x, the y-coordinate can be expressed as '1/a'.
Therefore, the distance formula becomes:
d = √((a - 0)² + (1/a - 0)²)
= √(a² + 1/a²)
To find the minimum distance, we need to minimize the expression inside the square root. Taking the derivative with respect to 'a' and setting it equal to zero will help us find the critical points.
Let f(a) = a² + 1/a². We can find the derivative of f(a) as follows:
f'(a) = 2a - 2/a³
Setting f'(a) = 0:
2a - 2/a³ = 0
Multiply through by a³ to clear the denominator:
2a⁴ - 2 = 0
Simplifying further:
a⁴ = 1
a = ±1
Since a must be positive, we can discard the negative solution.
Now, we have the x-coordinate of the point that is closest to the origin, which is a = 1.
To find the corresponding y-coordinate:
y = 1/a = 1/1 = 1
Therefore, the point on the hyperbola y = 1/x that is closest to the origin is (1, 1).