Text description - screencast for exercise 12

[The narrator reads out the onscreen text.]

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.

[The equation reads x squared plus (negative 2 plus 5) all squared, equals 5. So x squared plus 4x squared minus 20x plus 25 equals 5. So 5x squared minus 20x plus 20 equals 0. So x squared minus 4x plus 4 equals 0. So x squared minus 2 all squared equals 0. So x equals 2.]

NARRATOR: 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.

[An arrow points to the equations reading "5x squared minus 20x plus 20 equals 0. So x squared minus 4x plus 4 equals 0."]

NARRATOR: 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.