| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Circles |
| Type | Find parameter values for tangency using discriminant |
| Difficulty | Moderate -0.3 This is a standard multi-part circle question testing routine techniques: reading center/radius from equation, finding intersections by substitution, and using the discriminant condition for tangency. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
12 A circle has equation $( x - 2 ) ^ { 2 } + y ^ { 2 } = 20$.\\
(i) Write down the radius of the circle and the coordinates of its centre.\\
(ii) Find the points of intersection of the circle with the $y$-axis and sketch the circle.\\
(iii) Show that, where the line $y = 2 x + k$ intersects the circle,
$$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
(iv) Hence find the values of $k$ for which the line $y = 2 x + k$ is a tangent to the circle.
\section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
\hfill \mbox{\textit{OCR MEI C1 2012 Q12 [12]}}