June 2024 Paper 2 Q20
20 \(AB\) is the tangent to the circle \(\quad (x - 3)^2 + (y - 4)^2 = 29 \quad\) at the point \((5, 9)\)
Work out the area of triangle \(AOB\). [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{9 - 4}{5 - 3}\) or \(\dfrac{5}{2}\) | M1 | oe gradient of radius through (5, 9) |
| \(-1 \div\) their \(\dfrac{5}{2}\) or \(-\dfrac{2}{5}\) | M1 | oe gradient of \(AB\) may be seen in equation of \(AB\) eg \(y = -\dfrac{2}{5}x + c\) their \(\dfrac{5}{2}\) cannot be \(\sqrt{29}\) or 29 gradient of \(AB = -\dfrac{2}{5}\) with no incorrect working is M1M1 |
| \(\dfrac{y - 9}{0 - 5} =\) their \(-\dfrac{2}{5}\) or \((y =)\ 11\) or \(\dfrac{9 - 0}{5 - x} =\) their \(-\dfrac{2}{5}\) or \((x =)\ \dfrac{55}{2}\) | M1 | oe eg \(y = -\dfrac{2}{5} \times 0 + 11\) or \(0 = -\dfrac{2}{5} \times x + 11\) must have scored 2nd M1 or must have scored 1st M1 and their ‘perpendicular’ gradient must be either \(1 \div\) their \(\dfrac{5}{2}\) or \(-\) their \(\dfrac{5}{2}\) \((y =)\ 11\) or \((x =)\ \dfrac{55}{2}\) may be seen on diagram |
| \(\dfrac{y - 9}{0 - 5} =\) their \(-\dfrac{2}{5}\) or \((y =)\ 11\) and \(\dfrac{9 - 0}{5 - x} =\) their \(-\dfrac{2}{5}\) or \((x =)\ \dfrac{55}{2}\) | M1dep | oe eg \(y = -\dfrac{2}{5} \times 0 + 11\) and \(0 = -\dfrac{2}{5} \times x + 11\) dep on 3rd M1 \((y =)\ 11\) and \((x =)\ \dfrac{55}{2}\) may be seen on diagram |
| \(\dfrac{605}{4}\) or \(151\dfrac{1}{4}\) or 151.25 | A1 | oe value |
Additional guidance
| Ignore conversion or simplification attempt after correct answer seen | |
| 3rd M1 and 4th M1 If using the equation of \(AB\) in the form \(y = mx + c\) they must use their gradient and have either (i) a correct value for \(c\) or (ii) a correct calculation for \(c\) eg for (ii) if \(c\) should be \(-2\) for their gradient a correct calculation for \(c\) might be \(c = 1 - 3\) | |
| After M0M0 no other marks can be awarded | |
| \(\dfrac{9}{5}\) (gradient of radius) \(\quad -\dfrac{5}{9}\) (gradient of \(AB\)) \(y = -\dfrac{5}{9}x + \dfrac{106}{9}\) (correct equation of \(AB\) for their gradient) \(y = \dfrac{106}{9} \quad x = \dfrac{106}{5}\) (two correct values for their line) | M0M1 M1M1A0 |
| \(\dfrac{5}{2}\) (gradient of radius) \(\quad \dfrac{2}{5}\) (gradient of \(AB\)) \(y = \dfrac{2}{5}x + 7\) (correct equation of \(AB\) for their gradient) \(y = 7 \quad x = -\dfrac{35}{2}\) (two correct values for their line) | M1M0 M1M1A0 |