Moderate -0.3 This is a straightforward circle question requiring completion of the square to find centre/radius (routine), recognizing that maximum distance QS occurs along the diameter through Q (standard geometric insight), and finding a tangent equation perpendicular to a radius (standard technique). All steps are textbook exercises with no novel problem-solving required, making it slightly easier than average.
3. A circle \(C\) has equation
$$x ^ { 2 } - 22 x + y ^ { 2 } + 10 y + 46 = 0$$
a. Find
i. the coordinates of the centre \(A\) of the circle
ii. the radius of the circle.
Given that the points \(Q ( 5,3 )\) and \(S\) lie on \(C\) such that the distance \(Q S\) is greatest,
b. find an equation of tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are constants to be found.
\(3x - 4y - 103 = 0\) or any (non-zero) integer multiple of this. Accept terms in any order but have the "\(= 0\)"
B1
M1
For a full method to find the equation of a line e.g. attempts to find radius gradient and takes negative reciprocal and uses these to form the equation. E.g. \(y + 13 = \frac{3}{4}(x-17)\)
A1
**(a) (i)** | $A(11,-5)$ | A1 | M1 Attempts to complete the square on both $x$ and $y$ terms. Accept $(x \pm 11)^2 + (y \pm 5)^2 = \cdots$ or imply this mark for a centre of $(\pm 11, \pm 5)$. E.g. $(x \pm 11)^2 - 11^2 + (y \pm 5)^2 - 5^2 + 46 = 0$ gives $(x \pm 11)^2 + (y \pm 5)^2 = 100$. Accept without brackets. May be written $x = 11, y = -5$ |
**(a) (ii)** | $10$ | A1 | The M mark must have been awarded so it can be scored following a centre of $(\pm 11, \pm 5)$. Do not allow for $\sqrt{100}$ or $\pm 10$ |
**(b)** | $3x - 4y - 103 = 0$ or any (non-zero) integer multiple of this. Accept terms in any order but have the "$= 0$" | B1 | $S$ is $(17,-13)$ or $m_T = \frac{3}{4}$. Either identifies the correct point $S$ where $A$ is the mid-point of $QS$ or finds the correct gradient for the tangent using coordinates $(11,-5)$ and $(5,3)$ and takes negative reciprocal. E.g. $\left(\frac{5+x}{2}, \frac{3+y}{2}\right) = (11,-5) \Rightarrow \frac{5+x}{2} = 11, \frac{3+y}{2} = -5 \Rightarrow x = 17, y = -13$ or $m_{QA} = \frac{-5-3}{11-5} = -\frac{4}{3}$ and $m_T = \frac{3}{4}$ |
| --- | --- | --- |
| | M1 | For a full method to find the equation of a line e.g. attempts to find radius gradient and takes negative reciprocal and uses these to form the equation. E.g. $y + 13 = \frac{3}{4}(x-17)$ |
| | A1 | |
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3. A circle $C$ has equation
$$x ^ { 2 } - 22 x + y ^ { 2 } + 10 y + 46 = 0$$
a. Find\\
i. the coordinates of the centre $A$ of the circle\\
ii. the radius of the circle.
Given that the points $Q ( 5,3 )$ and $S$ lie on $C$ such that the distance $Q S$ is greatest,\\
b. find an equation of tangent to $C$ at $S$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are constants to be found.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q3 [6]}}