The line $y = 3x$ intersects the circle with equation $(x - 2)^2 + (y - 1)^2 = 25$ - Scottish Highers Maths - Question 3 - 2017

Expanding the equation:

$(x - 2)^2 + (3x - 1)^2 = 25$

$(x^2 - 4x + 4) + (9x^2 - 6x + 1) = 25$

Combining like terms:

$10x^2 - 10x + 5 = 25$

Subtracting 25 from both sides gives:

$10x^2 - 10x - 20 = 0$

Dividing through by 10:

$x^2 - x - 2 = 0$

Factoring gives:

$(x - 2)(x + 1) = 0$

Setting each factor to zero yields:

$x - 2 = 0 \Rightarrow x = 2$ $x + 1 = 0 \Rightarrow x = -1$

Using $y = 3x$ to find the corresponding y coordinates:

1. For $x = 2$: $y = 3(2) = 6$ Thus, one point of intersection is $(2, 6)$.

2. For $x = -1$: $y = 3(-1) = -3$ Thus, the other point of intersection is $(-1, -3)$.