NARRATOR: Exercise 12.
Show that the line y = –2x + 5
is tangent to the circle x squared + y squared = 5.
If the line is tangent to the circle,
then there is only one point of intersection
between the line and the circle.
The point or points of intersection can be found
by substituting the linear equation into the equation for the circle,
giving the following.
Expanding –2x + 5 all squared and then collecting like terms
gives a quadratic expression.
Dividing through by 5, then factorising and then solving
shows that x = 2 is the only solution to this equation.
So since there is only one point of intersection
between the line and the circle, at the point where x = 2,
the line must be tangent to the circle.
At this point of the process, we could also have shown that
the discriminant of this quadratic is equal to 0,
and hence there is only one solution to this equation
that gives the points of intersection,
and therefore only one point of intersection.
But in this case, it was very simple to solve this quadratic
once we reach this point.