The circle C has equation $x^2 + y^2 - 10x + 4y + 11 = 0$ (a) Find (i) the coordinates of the centre of C, (ii) the exact radius of C, giving your answer as a simplified surd - Edexcel - A-Level Maths Pure - Question 8 - 2021 - Paper 1

The radius $r$ of the circle is defined as the square root of the constant on the right side of the equation: $r = ext{sqrt{18}}$ This gives: $r = ext{sqrt{9 imes 2}} = 3 ext{sqrt{2}}$ Thus, the radius of circle C is $3 ext{sqrt{2}}$.

To find the values of k, we substitute $y = 3x + k$ into the circle's equation:

1. Substitute:

   • Substitute for $y$ in the circle's equation: $x^2 + (3x + k)^2 - 10x + 4(3x + k) + 11 = 0$

2. Expand:

   • Expanding gives: $x^2 + (9x^2 + 6kx + k^2) - 10x + 12x + 4k + 11 = 0$ Which simplifies to: $10x^2 + (6k + 2)x + (k^2 + 4k + 11) = 0$

3. Set the discriminant to zero (since l is a tangent):

   • For the quadratic in $x$ to have exactly one solution (tangent), the discriminant must be zero: $b^2 - 4ac = 0$ Here, $a = 10$, $b = (6k + 2)$, and $c = (k^2 + 4k + 11)$: $(6k + 2)^2 - 4(10)(k^2 + 4k + 11) = 0$ This expands and simplifies accordingly.

4. Solve for k:

   • After completing the algebraic manipulation, we find: $k = 17 ext{ or } k = -6 ext{sqrt{5}}$ Thus, the possible values of k are $17$ and $-6 ext{sqrt{5}}$.