Higher June 2025 Paper 2 Q21
21 A circle with centre \(O\) and radius \(\sqrt{45}\) has a tangent drawn at point \(P(6, -3)\)
Not drawn accurately
(a) Write down the equation of the circle. [1 mark]
(b) Work out the equation of the tangent.
Give your answer in the form \(y = mx + c\) [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(x^2 + y^2 = 45\) | B1 | oe accept \(\sqrt{45}^{\,2}\) or \(\sqrt{45^2}\) or \((3\sqrt{5})^2\) or \([6.7, 6.71]^2\) for 45 |
Additional guidance
Accept \((x - 0)^2\) for \(x^2\) and \((y - 0)^2\) for \(y^2\)
Do not accept eg \(a^2 + b^2 = 45\)
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{-3 - 0}{6 - 0}\) or \(-\dfrac{1}{2}\) | M1 | oe gradient of \(OP\) eg \(\dfrac{0 - (-3)}{0 - 6}\) may be embedded in an equation |
| \(-1 \div\) their \(-\dfrac{1}{2}\) or 2 | M1 | oe gradient of tangent may be seen in equation of tangent their \(-\dfrac{1}{2}\) cannot be from using 45 gradient of tangent \(= 2\) with no incorrect working is M1M1 |
| \(y - (-3) =\) their \(2(x - 6)\) or \(-3 =\) their \(2 \times 6 + c\) or \(c = -15\) | M1dep | oe equation eg \(y + 3 = 2x - 12\) dep on 2nd M1 |
| \(y = 2x - 15\) | A1 | oe in form \(y = mx + c\) eg \(y = \dfrac{6}{3}x - 15\) accept \(y = -15 + 2x\) |
Additional guidance
| (gradient of \(OP =\)) \(\dfrac{1}{2}\) (gradient of tangent \(=\)) \(-2\) \(y + 3 = -2(x - 6)\) \(y = -2x + 9\) | M0 M1 M1dep A0 |
| (gradient of \(OP =\)) \(\sqrt{45}\) (may be seen as \(3\sqrt{5}\) or [6.7, 6.71]) (gradient of tangent \(=\)) \(-\dfrac{1}{\sqrt{45}}\) | M0 M0 |