How do you fInd the equation(s) of the tangent line(s) at the points) on the graph of the equation #y^2 -xy-10=0#, where x=3?

Question

Cameron Alexander · Accepted Answer

Please see the explanation.

When x = 3, the equation becomes:

#y^2 - 3y - 10 = 0#

This factors into:

#(y + 2)(y - 5) = 0#

The roots are:

#y = -2 and y = 5#

The points of tangency are #(3, -2) and (3, 5)#

Now, use implicit differentiation to compute the first derivative.

#2ydy/dx - y - xdy/dx = 0#

#(2y - x)dy/dx = y#

#dy/dx = y/(2y -x)#

At the point #(3, -2)#, the slope of the tangent line is:

#m = (-2)/(2(-2) - 3) = 2/7#

#-2 = 2/7(3) + b#

#b = -20/7#

The equation of the tangent line is:

#y = 2/7x - 20/7#

At the point #(3, 5)#, the slope of the tangent line is:

#m = 5/(2(5) - 3) = 5/7#

#5 = 5/7(3) + b#

#b = 20/7#

The equation of the tangent line is:

#y = 5/7x + 20/7#

The following graph shows the curve, the points of tangency, and the tangent lines.

Adam Adkins · Answer

undefined To find the equation(s) of the tangent line(s) at the point(s) on the graph of the equation y^2 - xy - 10 = 0, where x = 3, we can follow these steps:

1. Differentiate the given equation implicitly with respect to x.
2. Substitute the value x = 3 into the resulting derivative equation.
3. Solve the resulting equation for y to find the corresponding y-coordinate(s) of the point(s) of tangency.
4. Substitute the values of x and y into the equation y = mx + c, where m represents the slope of the tangent line(s).
5. Solve for c, the y-intercept of the tangent line(s), using the point-slope form of the equation.
6. Write the equation(s) of the tangent line(s) in the form y = mx + c, using the values of m and c obtained.

Please note that the solution may yield one or two tangent lines depending on the nature of the graph.