# Determine equations of all lines

• Kazane
In summary, the task is to find the equations of all lines that are tangent to the curve y=x^2/(x-1) and pass through the point (2,0). The first step is to find the derivative of the curve, which is y'=x(x-2)/(x-1)^2. Then, using the slope formula, we can set the slope of the tangent line equal to the slope of the line passing through the point (2,0) and a general point (a,f(a)) on the curve. This will result in an equation that can be solved for a, giving us the x-coordinate of the point(s) of tangency. Finally, using the x-coordinate, we can find the y
Kazane

## Homework Statement

Determine equations of all lines the are tangent to y=x^2/(x-1) and that pass through the point (2,0).

## The Attempt at a Solution

First I found the y' which I came up with y'=x(x-2)/(x-1)^2

next, let (a,f(a)) and put a to y'
f'(a)=a(a-2)/(a-1)^2 (slope of tangent lines at x=a) ->①
also this slope is given by f(a)-0/a-2 -> ②
①=② then solve for a, but I stuck for calculating this...

Am I on the right truck ?? can anyone help me, please??

You have the x,y co-ordinates of a general point on the curve, you have the slope of the tangent through that point, get the equation for that line. When you have done that I expect you can do the rest too.

What is the slope of a line passing through points (x1, y1) and (x2, y2) ?

Then:
Let (x2, y2) = (a, f(a)), and (x1, y1) = (2, 0). You know the slope of the line passing through these two points must be f ' (a) , if the line is to be tangent to the curve y=x2/(x-1) at x = a.

## 1. How do you determine the equation of a line given two points?

To determine the equation of a line, you need to use the slope-intercept formula: y = mx + b. First, calculate the slope (m) using the formula (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. Then, substitute the slope and the coordinates of one point into the formula to solve for b. This will give you the equation of the line in the form of y = mx + b.

## 2. What is the difference between standard form and slope-intercept form for equations of lines?

The standard form of a line is written as Ax + By = C, where A, B, and C are constants. This form is useful for graphing and finding the x- and y-intercepts. The slope-intercept form, y = mx + b, is more commonly used because it shows the slope (m) and y-intercept (b) of the line.

## 3. Can you determine the equation of a line given only the slope?

Yes, you can determine the equation of a line given only the slope. You will still need to know the coordinates of at least one point on the line, which can be used to solve for the y-intercept (b) in the slope-intercept form.

## 4. How do you determine the equation of a line parallel or perpendicular to a given line?

To determine the equation of a line parallel or perpendicular to a given line, you need to use the fact that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. Use the slope of the given line to find the slope of the parallel or perpendicular line, and then use the point-slope form (y - y1) = m(x - x1) with the slope and a point on the new line to find its equation.

## 5. Are there any other forms or methods for determining the equation of a line?

Yes, there are other forms and methods for determining the equation of a line. Some other common forms include point-slope form, two-point form, and intercept form. Additionally, you can use linear regression or other statistical methods to find the equation of a line that best fits a set of data points.

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