Value of when a line touches a parabola at one point.

In summary, in order for the line y = mx to touch the parabola y = (x - 1)2 + 1 at just one point, the value of m must be such that the quadratic equation x^2-(2+m)x+2=0 has only one solution, which happens when b^2-4ac=0.
  • #1
mattstjean
14
0
The Question asks:
Find the value of m > 0 for which the line y = mx touches the parabola y = (x - 1)2 + 1 at just one point.

So far what I've done...

I know that if the line touches the parabola at one point, it is tangential. So I put together the 2 equations in order to find the intersection (or touch):

mx=(x-1)2+1
0=x2-2x-mx+2
0=x2-(2+m)x+2

That's as far as I've gotten. Someone said I need to find the restriction on 'm' so that the equation only has one root (just touches, as opposed to a full intersection through 2 points), but I don't know what to do...

Any help would be appreciated.
 
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  • #2
mattstjean said:
The Question asks:
Find the value of m > 0 for which the line y = mx touches the parabola y = (x - 1)2 + 1 at just one point.

So far what I've done...

I know that if the line touches the parabola at one point, it is tangential. So I put together the 2 equations in order to find the intersection (or touch):

mx=(x-1)2+1
0=x2-2x-mx+2
0=x2-(2+m)x+2

That's as far as I've gotten. Someone said I need to find the restriction on 'm' so that the equation only has one root (just touches, as opposed to a full intersection through 2 points), but I don't know what to do...

Any help would be appreciated.

HINT: Given a quadratic equation [tex]ax^2+bx+c=0[/tex], its solutions are given by

[tex]x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}[/tex]

What should the value of m be so that there is only one solution?
 
  • #3
danago said:
HINT: Given a quadratic equation [tex]ax^2+bx+c=0[/tex], its solutions are given by

[tex]x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}[/tex]

What should the value of m be so that there is only one solution?

The quadratic formula was my first guess however, since it has two variables: m and x...I didn't know how to use it properly.
 
  • #4
mattstjean said:
The quadratic formula was my first guess however, since it has two variables: m and x...I didn't know how to use it properly.

You have x^2-(2+m)x+2=0. A general quadratic equation has the form a*x^2+b*x+c=0. a, b and c depend only on m. They don't depend on x. Just identify a, b and c.
 
  • #5
danago said:
HINT: Given a quadratic equation [tex]ax^2+bx+c=0[/tex], its solutions are given by

[tex]x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}[/tex]

What should the value of m be so that there is only one solution?

Is it when b2-4ac = 0 ?
 
  • #6
mattstjean said:
Is it when b2-4ac = 0 ?

Yes.
 

Related to Value of when a line touches a parabola at one point.

1. What is the significance of a line touching a parabola at one point?

When a line touches a parabola at one point, it represents a unique solution to the system of equations created by the line and the parabola. This point of intersection is known as the point of tangency, and it can provide valuable information about the relationship between the two curves.

2. How can the point of tangency be used to find the value of the line?

The coordinates of the point of tangency can be used to find the slope of the line, which can then be used to determine the equation of the line. This can be done by finding the derivative of the parabola at the point of tangency, which will be equal to the slope of the line.

3. Can a line touch a parabola at more than one point?

Yes, a line can touch a parabola at more than one point. This will result in a system of equations with multiple solutions, which can be solved using various algebraic methods. However, in most cases, a line and a parabola will intersect at only one point.

4. What does it mean if a line does not touch a parabola at any point?

If a line does not touch a parabola at any point, it means that the two curves do not intersect and there is no solution to the system of equations created by them. This can happen when the line and the parabola are parallel or when the line is outside of the parabola's range of values.

5. How does the position of the line and parabola affect their point of tangency?

The position of the line and parabola relative to each other will determine the coordinates of their point of tangency. If the line is above the parabola, the point of tangency will have a positive y-coordinate, and if the line is below the parabola, the point of tangency will have a negative y-coordinate. Similarly, the x-coordinate of the point of tangency will be affected by the position of the line and parabola in the x-direction.

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