 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/10/solution

1

centred at

(2, 1)

. A line through the origin with equation

k x - y = 0

intersects it precisely when the perpendicular distance from that centre is at most the radius. The distance from a point to a line gives

\frac{∣2 k - 1∣}{k ^{2} + 1} \leq 1 ⟺ (2 k - 1)^{2} \leq k^{2} + 1 ⟺ k (3 k - 4) \leq 0.

Hence the full parameter range is

0 \leq k \leq \frac{4}{3} .

The two endpoint lines are tangent, at

(2, 0)

and

(6/5, 8/5)

respectively; interior values give two real intersections. Both endpoints are therefore included.

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