Equations of the tangent lines to hyperbola xy=1 that pass through point (-1,1)

I know the graph of y=1/x but not sure about the tangent lines at given point. Are the lines tangent to (1,1) and (-1,-1)?

You're correct, the graph is y=1/x, so find y' to get the slope at any point on the graph (except the origin). I only found one tangent line through (-1,1), I'm not sure what other point the question has in mind.

To find the equation of the tangent lines to the hyperbola xy = 1 that pass through the point (-1, 1), we first need to find the derivative of the hyperbola. Using implicit differentiation, we have:

x(dy/dx) + y = 0

Solving for dy/dx, we get:

dy/dx = -y/x

Now, let's consider a tangent line that passes through the point (-1, 1). We need to find a point (x_1, y_1) on the hyperbola such that the slope between (-1, 1) and (x_1, y_1) equals the value of the derivative at (x_1, y_1).

Using the point-slope form of a line, we have:

(y - 1) / (x + 1) = -(y_1) / (x_1)

But since (x_1, y_1) is a point on the hyperbola, we have x_1 * y_1 = 1.

Now we have a system of equations:

(y - 1) / (x + 1) = -(y_1) / (x_1)
x_1 * y_1 = 1

Solve this system to find (x_1, y_1). We'll first express y_1 in terms of x_1:

y_1 = 1 / x_1

Now substitute this into the first equation:

(y - 1) / (x + 1) = -(1 / x_1) / (x_1)

Multiply both sides by x_1(x + 1) to eliminate the denominators:

x_1(y - 1) = -(x + 1)

Now substitute y = 1/x back in:

x_1(1/x - 1) = -(x + 1)

Simplify and solve for x:

x^2 - x_1^2 = -x_1^2 - x_1

x^2 = -x_1

But since x^2 is always non-negative, there is no solution for x_1, and therefore no tangent line that passes through the point (-1, 1). The point (-1, 1) does not lie on the hyperbola, so there is no tangent line from the point to the hyperbola.

To find the slope of the tangent line at any point on the graph, we need to differentiate the equation xy = 1 with respect to x.

Differentiating with respect to x:
d(xy)/dx = d(1)/dx

Using the product rule:
(x * dy/dx) + (y * dx/dx) = 0

Simplifying:
x * dy/dx + y = 0

Rearranging:
dy/dx = -y/x

Now, substitute the given point (-1, 1) into the equation y = 1/x to find the y-coordinate at that point:

y = 1/(-1) = -1

Therefore, at the point (-1, 1), the y-coordinate is -1.

Now, substitute the values of x and y into the equation dy/dx = -y/x:

dy/dx = -(-1)/(-1) = -1

So, the slope of the tangent line at the point (-1, 1) is -1.

To find the equation of the tangent line, we use the point-slope form of a linear equation:

y - y1 = m(x - x1)

Substituting the values (-1, 1) and -1 for x1, y1, and m, respectively:

y - 1 = -1(x - (-1))

Simplifying:
y - 1 = -1(x + 1)
y - 1 = -x - 1

Rearranging the equation in slope-intercept form gives the equation of the tangent line passing through the point (-1, 1):

y = -x

So, the equation of the tangent line to the hyperbola xy = 1 that passes through the point (-1, 1) is y = -x.

To find the equations of the tangent lines to the hyperbola xy=1 that pass through the point (-1,1), we can follow these steps:

Step 1: Differentiate the equation xy=1 with respect to x to find the derivative dy/dx.
To do this, we will use implicit differentiation. Differentiating both sides of the equation with respect to x, we get:
(1 * dy/dx) + (x * d/dx[y]) = 0
Simplifying, we get:
dy/dx + x * d/dx[y] = 0

Step 2: Solve for d/dx[y] to obtain the derivative of y with respect to x.
Rearranging the equation from step 1, we get:
d/dx[y] = -dy/dx / x

Step 3: Substitute the given point (-1,1) into the equation xy=1 to find the corresponding coordinates (x,y).
Plugging in the values of x and y, we have:
(-1)(1) = 1
-1 = 1
This is not true, so (-1,1) does not lie on the hyperbola xy=1, which means there are no tangent lines through (-1,1) on the hyperbola.

Moreover, the lines tangent to the hyperbola xy=1 will not pass through the points (1,1) and (-1,-1). You may have mistaken them for the points of intersection between the hyperbola and the asymptotes. The asymptotes of the hyperbola intersect at the origin (0,0), not (1,1) and (-1,-1).

Therefore, there are no tangent lines to the hyperbola xy=1 that pass through the point (-1,1).