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 equation of the circle is now of the form:

$(x - h)^2 + (y - k)^2 = r^2$

where (h, k) is the centre and r is the radius. Comparing, we find:

$r^2 = 18$

Thus,

oot{2}}{1}=rac{3}{ oot{2}}$$ The radius of C as a simplified surd is $rac{3}{ oot{2}}$.

Given that the line $l$ is a tangent to C, we substitute

$y = 3x + k$ into the circle's equation:

$x^2 + (3x + k)^2 - 10x + 4(3x + k) + 11 = 0$

Expanding this:

$x^2 + (9x^2 + 6kx + k^2) - 10x + 12x + 4k + 11 = 0$

Combining like terms, we have:

$10x^2 + (6k + 2)x + (k^2 + 4k + 11) = 0$

For the line to be tangent to the circle, the discriminant must equal zero:

$b^2 - 4ac = 0$

Here, $a = 10$, $b = (6k + 2)$, and $c = (k^2 + 4k + 11)$:

$(6k + 2)^2 - 4(10)(k^2 + 4k + 11) = 0$

Expanding gives:

$36k^2 + 24k + 4 - 40k^2 - 160k - 440 = 0$

Combining terms results in:

$-4k^2 - 136k - 436 = 0$

Simplifying further yields:

$4k^2 + 136k + 436 = 0$

Dividing through by 4:

$k^2 + 34k + 109 = 0$

Using the quadratic formula gives:

oot{(34^2 - 4 imes 1 imes 109)}}{2 imes 1}$$ Evaluating the roots leads to: $$k = -17 ext{±} rac{ oot{76}}{2} = -17 ext{±} rac{4 oot{19}}{2}$$ Thus, the possible values of k are: $$k = -17 + oot{76}$$ $$k = -17 - oot{76}$$