Find m if the line y=mx−2 intersects y=x^2 in just one point.
m is equal to plus or minus sqrt8
To find the value of m for which the line y = mx - 2 intersects the parabola y = x^2 in just one point, we can equate the two equations and solve for x.
Substituting y = mx - 2 into the equation of the parabola, we have:
mx - 2 = x^2
Rearranging the equation, we get:
x^2 - mx + 2 = 0
For the line and parabola to intersect in just one point, this quadratic equation must have only one solution. This means that the discriminant (b^2 - 4ac) of the quadratic equation must be equal to zero.
The discriminant is:
b^2 - 4ac = (-m)^2 - 4(1)(2) = m^2 - 8
Setting the discriminant equal to zero, we have:
m^2 - 8 = 0
Solving for m, we find:
m^2 = 8
Taking the square root of both sides, we get:
m = ±√8
So, the two possible values of m for which the line and the parabola intersect in just one point are:
m = √8
m = -√8
To find the value of m such that the line y = mx - 2 intersects the curve y = x^2 at exactly one point, we need to set up an equation and solve for m.
Step 1: Set the equations equal to each other.
Since the line and the curve intersect at exactly one point, their respective y-values must be equal. Therefore, we set mx - 2 equal to x^2:
mx - 2 = x^2
Step 2: Rearrange the equation into a quadratic form.
To solve for m, we need to rewrite the equation in the form of a quadratic equation, ax^2 + bx + c = 0. Let's move all terms to one side to have a quadratic equation:
x^2 - mx + 2 = 0
Step 3: Determine the discriminant.
To find the number of solutions for the quadratic equation, we can use the discriminant, which is the part of the quadratic formula under the square root sign. The discriminant is given by:
D = b^2 - 4ac
In our case, a=1, b=-m, and c=2. Substituting these values into the discriminant formula, we have:
D = (-m)^2 - 4(1)(2)
D = m^2 - 8
Step 4: Find the condition for a single intersection point.
For a quadratic equation, if the discriminant is equal to zero, it means there is only one solution. Therefore, we set the discriminant D equal to zero:
m^2 - 8 = 0
Step 5: Solve for m.
To find m, we solve the quadratic equation:
m^2 - 8 = 0
Factoring the equation, we have:
(m + √8)(m - √8) = 0
Setting each factor equal to zero, we get:
m + √8 = 0 or m - √8 = 0
Solving for m in each case:
m = -√8 or m = √8
So, the values of m that satisfy the condition for the line y = mx - 2 to intersect the curve y = x^2 at exactly one point are m = -√8 and m = √8.
you want the solution to
mx-2 = x^2
to have a zero discriminant. So, since
x^2 - mx + 2 = 0
has discriminant of
m^2 - 8
I guess that tells you what m is, right? Take a look at
http://www.wolframalpha.com/input/?i=plot+y%3Dx^2,+y%3D%E2%88%9A8+x-2,+y%3D-%E2%88%9A8+x-2