Find the Tangent Lines of y=x/(x+1) Through (1,2)

• davedave
In summary, the conversation discusses finding the equations of two tangent lines to a curve that passes through a given point. The suggested method involves sketching the graph, using the derivative and the formula of a line, and checking for the condition of the quadratic equation having two equal roots. Additionally, the restrictions for the line to pass through the given point and the use of Winplot are mentioned.
davedave
Can someone help me with this problem?

Consider the curve defined by y=x/(x+1). (1,2) is a point NOT on the curve.
Find the equations of the two tangent lines to the curve passing through the point
(1,2).

There are some ways to attack this problem but my first instinct is to sketch or print the graph then mark the point (1,2) then take a ruler and carefully draw the tangent lines touching one point on each side of the graph. Then take the derivative around those points and use the formula of a line (y=mx+c) and check to see if the gradients run through the points.

This is perhaps tedious to say the least but I don't know how else to tackle the problem without any more information.

btw winplot should help with this.

hihi davedave!

Write y = ax + b for a typical line.

What is the restriction on a and b for it to pass through (1,2)?

Now find the equation for the points of intersection of that line with y=x/(x+1).

That should be a quadratic equation.

What is the condition for that equation to have two equal roots?

1. What is the equation for finding the tangent lines of y=x/(x+1) through (1,2)?

The equation for finding the tangent lines of y=x/(x+1) through (1,2) is y = (x+2)/(x+1).

2. How do you determine the slope of the tangent lines?

The slope of the tangent lines can be determined by taking the derivative of the function y=x/(x+1) and plugging in the x-value of the given point (1,2). The resulting value is the slope of the tangent lines.

3. Can there be more than one tangent line through the point (1,2)?

Yes, there can be more than one tangent line through the point (1,2). This is because the given point lies on a curve and there can be multiple lines that are tangent to the curve at that point.

4. How do you graph the tangent lines of y=x/(x+1) through (1,2)?

To graph the tangent lines, you can plot the given point (1,2) on the graph and then use the slope you calculated earlier to find a second point on the line. Connect the two points to draw the tangent line. Repeat this process for any other tangent lines that pass through the point (1,2).

5. Can you use calculus to find the tangent lines of a curve at any given point?

Yes, calculus can be used to find the tangent lines of a curve at any given point. This is because the derivative of a function at a specific point gives the slope of the tangent line to the curve at that point.

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