Find m if the line y=mx−2 intersects y=x^2 in just one point.

we have that
y=mx−2
y=x²

if the line intersects a the parabola in one point
so
mx-2=x²-------> x²-mx+2=0
Find the discriminant D=b²-4ac
a=1
b=-m
c=2
D=m²-8
we know that

If D is negative ------> Then the square root is imaginary and there is no solution
If D is 0 then there is exactly one solution
if D is positive there are two solution
therefore
equals D to zero
m²-8=0-------> m=(+/-)√8

the answer is
m=(+/-)√8

see the attached figure
m has two solutions

Answer Link

MrRoyal MrRoyal · Answer 2 · 2022-03-22T11:54:31+01:00

The value of m if the line y=mx−2 intersects y=x^2 in just one point is [tex]m = \pm \sqrt 8[/tex]

The equations are given as:

y=mx−2

y=x²

When the line and the curve intersects, we have:

y = y

So, the equation becomes

mx-2=x²

Rewrite the equation as:

x²-mx+2=0

When they have just one point of intersection, then we have the following relationship

b² = 4ac

Where

a=1

b=-m

c=2

So, we have:

(-m)²= 4 * 1 * 2

Evaluate the exponent and the product

m² = 8

Take the square root of both sides

[tex]m = \pm \sqrt 8[/tex]

Hence, the value of m if the line y=mx−2 intersects y=x^2 in just one point is [tex]m = \pm \sqrt 8[/tex]