The straight line x + y = c, where c is some number, touches the hyperbola y = 1 / x at a point

The straight line x + y = c, where c is some number, touches the hyperbola y = 1 / x at a point with positive coordinates. Find c.

We have the function y = 1 / x.

We also have a straight line x + y = c, which touches the hyperbola at a point with positive coordinates. Find c.

Let’s write the equation of a straight line in a different form:

y = c – x;

And we write down the equation of the tangent line at the point x0:

y = y ‘(x0) * (x – x0) + y (x0);

The coefficient of the variable is minus one, at the same time it is the value of the derivative at the point x0:

y ‘(x0) = -1;

y ‘(x) = -1 / x ^ 2;

-1 / x0 ^ 2 = -1;

x0 ^ 2 = 1;

x0 = 1, since positive coordinates of the point are indicated in the problem statement.

y (x0) = 1/1 = 1;

Let’s write the equation of the tangent line:

y = -1 * (x – 1) + 1;

y = -x + 1 + 1;

y = -x + 2.

c = 2.