Hello, I need help finding the intersection points B and C of the tangent A with for horizontal axis B and the vertical axis C.
hyperbola: f (x) = 1 / x
A = (1, 1)
Thanks for your help .
f' = -1/x^2
at (1,1) y' = -1
so, you have a point and a slope.
y-1 = -1(x-1)
you can probably find the intercepts from here, no?
B = (2, 0) and C = (0, 2) but we have explained how to find this result.
To find the intersection points B and C, we need to find the x and y coordinates where the tangent line intersects the x-axis and y-axis, respectively.
Given the equation of the hyperbola is f(x) = 1/x, we can find the derivative of f(x) to find the slope of the tangent line.
Let's begin by finding the derivative of f(x):
f'(x) = -1/x^2
Now, we can find the slope of the tangent line at point A by substituting x=1 into the derivative:
m = f'(1) = -1/1^2 = -1
So, the slope of the tangent line is -1.
Using the point-slope form of a line, we can write the equation of the tangent line:
y - y₁ = m(x - x₁)
Substituting the values of point A (1, 1) and the slope -1:
y - 1 = -1(x - 1)
Simplifying the equation:
y - 1 = -x + 1
y = -x + 2
Now, to find the x-intercept (point B), we set y = 0 in the equation of the tangent line:
0 = -x + 2
x = 2
So, B = (2, 0).
To find the y-intercept (point C), we set x = 0 in the equation of the tangent line:
y = -0 + 2
y = 2
So, C = (0, 2).
Therefore, the intersection points B and C of the tangent line with the x-axis and y-axis, respectively, are B = (2, 0) and C = (0, 2).