Prove algebraically that the straight line with equation $x - 2y = 10$ is a tangent to the circle with equation $x^2 + y^2 = 20$. - Edexcel - GCSE Maths - Question 19 - 2017 - Paper 3

Now expand the equation:

$x^2 + \frac{(x - 10)^2}{4} = 20$

This simplifies to:

$x^2 + \frac{x^2 - 20x + 100}{4} = 20$

Multiply through by 4 to eliminate the fraction:

$4x^2 + (x^2 - 20x + 100) = 80$

Now combine like terms:

$5x^2 - 20x + 20 = 0$

To determine if the line is tangent, we need to check the discriminant of the quadratic equation:

$D = b^2 - 4ac$

For our equation, $a=5$, $b=-20$, and $c=20$:

$D = (-20)^2 - 4(5)(20)$ => $D = 400 - 400 = 0$

Since the discriminant is zero, this means that there is exactly one solution, indicating that the line is tangent to the circle.

Therefore, we have algebraically proven that the line $x - 2y = 10$ is tangent to the circle $x^2 + y^2 = 20$ since it intersects at exactly one point.