Coordinates from geometric constraints - Straight Lines & Coordinate Geometry

CAIE P1 2021 June Q6

4 marks Standard +0.3

6 Points $A$ and $B$ have coordinates $( 8,3 )$ and $( p , q )$ respectively. The equation of the perpendicular bisector of $A B$ is $y = - 2 x + 4$. Find the values of $p$ and $q$.

CAIE P1 2009 June Q8

7 marks Standard +0.8

8 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_599_716_1071_717} The diagram shows points $A , B$ and $C$ lying on the line $2 y = x + 4$. The point $A$ lies on the $y$-axis and $A B = B C$. The line from $D ( 10 , - 3 )$ to $B$ is perpendicular to $A C$. Calculate the coordinates of $B$ and $C$.

CAIE P1 2010 June Q8

10 marks Standard +0.3

8 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_625_547_1489_797} The diagram shows a triangle $A B C$ in which $A$ is $( 3 , - 2 )$ and $B$ is $( 15,22 )$. The gradients of $A B , A C$ and $B C$ are $2 m , - 2 m$ and $m$ respectively, where $m$ is a positive constant.

Find the gradient of $A B$ and deduce the value of $m$.
Find the coordinates of $C$. The perpendicular bisector of $A B$ meets $B C$ at $D$.
Find the coordinates of $D$.

CAIE P1 2011 June Q3

5 marks Standard +0.3

3 The line $\frac { x } { a } + \frac { y } { b } = 1$, where $a$ and $b$ are positive constants, meets the $x$-axis at $P$ and the $y$-axis at $Q$. Given that $P Q = \sqrt { } ( 45 )$ and that the gradient of the line $P Q$ is $- \frac { 1 } { 2 }$, find the values of $a$ and $b$.

CAIE P1 2016 June Q8

8 marks Moderate -0.8

8 Three points have coordinates $A ( 0,7 ) , B ( 8,3 )$ and $C ( 3 k , k )$. Find the value of the constant $k$ for which

$C$ lies on the line that passes through $A$ and $B$,
$C$ lies on the perpendicular bisector of $A B$.

CAIE P1 2007 November Q6

7 marks Standard +0.3

6 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-2_627_748_1685_699} The three points $A ( 3,8 ) , B ( 6,2 )$ and $C ( 10,2 )$ are shown in the diagram. The point $D$ is such that the line $D A$ is perpendicular to $A B$ and $D C$ is parallel to $A B$. Calculate the coordinates of $D$.

CAIE P1 2011 November Q9

9 marks Standard +0.8

9 \includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-4_767_993_255_575} The diagram shows a quadrilateral $A B C D$ in which the point $A$ is ( $- 1 , - 1$ ), the point $B$ is ( 3,6 ) and the point $C$ is (9,4). The diagonals $A C$ and $B D$ intersect at $M$. Angle $B M A = 90 ^ { \circ }$ and $B M = M D$. Calculate

the coordinates of $M$ and $D$,
the ratio $A M : M C$.

CAIE P1 2016 November Q5

6 marks Standard +0.3

5 The line $\frac { x } { a } + \frac { y } { b } = 1$, where $a$ and $b$ are positive constants, intersects the $x$ - and $y$-axes at the points $A$ and $B$ respectively. The mid-point of $A B$ lies on the line $2 x + y = 10$ and the distance $A B = 10$. Find the values of $a$ and $b$.

Edexcel C12 2014 June Q5

7 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-07_953_929_219_422} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Diagram not drawn to scale Figure 3 shows the points $P , Q$ and $R$. Points $P$ and $Q$ have coordinates ( $- 1,4$ ) and ( 4,7 ) respectively.

Find an equation for the straight line passing through points $P$ and $Q$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers. The point $R$ has coordinates ( $p , - 3$ ), where $p$ is a positive constant. Given that angle $Q P R = 90 ^ { \circ }$,
find the value of $p$.

Edexcel C12 2017 October Q10

9 marks Moderate -0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-24_863_929_255_511} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Diagram NOT drawn to scale The points $A ( 7 , - 3 ) , B ( 7,20 )$ and $C ( p , q )$ form the vertices of a triangle $A B C$, as shown in Figure 4. The point $D ( 10,5 )$ is the midpoint of $A C$.

Find the value of $p$ and the value of $q$. The line $l$ passes through $D$ and is perpendicular to $A C$.
Find an equation for $l$, in the form $a x + b y = c$, where $a$, $b$ and $c$ are integers. Given that the line $l$ intersects $A B$ at $E$,
find the exact coordinates of $E$.

Edexcel C12 2018 October Q7

8 marks Moderate -0.8

7. The point $A$ has coordinates $( - 1,5 )$ and the point $B$ has coordinates $( 4,1 )$. The line $l$ passes through the points $A$ and $B$.

Find the gradient of $l$.
Find an equation for $l$, giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers. The point $M$ is the midpoint of $A B$. The point $C$ has coordinates $( 5 , k )$ where $k$ is a constant.
Given that the distance from $M$ to $C$ is $\sqrt { 13 }$
find the exact possible values of the constant $k$.

Edexcel C1 2005 January Q8

9 marks Moderate -0.8

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-14_687_1196_280_388}

\end{figure} The points $A ( 1,7 ) , B ( 20,7 )$ and $C ( p , q )$ form the vertices of a triangle $A B C$, as shown in Figure 2. The point $D ( 8,2 )$ is the mid-point of $A C$.

Find the value of $p$ and the value of $q$. The line $l$, which passes through $D$ and is perpendicular to $A C$, intersects $A B$ at $E$.
Find an equation for $l$, in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
Find the exact $x$-coordinate of $E$.

Edexcel C1 2014 June Q7

9 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-10_869_1073_267_440} \captionsetup{labelformat=empty} \caption{Diagram NOT to scale}

\end{figure} Figure 2 Figure 2 shows a right angled triangle $L M N$. The points $L$ and $M$ have coordinates ( $- 1,2$ ) and ( $7 , - 4$ ) respectively.

Find an equation for the straight line passing through the points $L$ and $M$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers. Given that the coordinates of point $N$ are ( $16 , p$ ), where $p$ is a constant, and angle $L M N = 90 ^ { \circ }$,
find the value of $p$. Given that there is a point $K$ such that the points $L , M , N$, and $K$ form a rectangle,
find the $y$ coordinate of $K$.

Edexcel FP1 2014 June Q8

11 marks Standard +0.8

8. The points $P \left( 4 k ^ { 2 } , 8 k \right)$ and $Q \left( k ^ { 2 } , 4 k \right)$, where $k$ is a constant, lie on the parabola $C$ with equation $y ^ { 2 } = 16 x$. The straight line $l _ { 1 }$ passes through the points $P$ and $Q$.

Show that an equation of the line $l _ { 1 }$ is given by $$3 k y - 4 x = 8 k ^ { 2 }$$ The line $l _ { 2 }$ is perpendicular to the line $l _ { 1 }$ and passes through the focus of the parabola $C$. The line $l _ { 2 }$ meets the directrix of $C$ at the point $R$.
Find, in terms of $k$, the $y$ coordinate of the point $R$.

OCR MEI C1 Q2

2 marks Easy -1.2

2 A line has gradient 3 and passes through the point $( 1 , - 5 )$. The point $( 5 , k )$ is on this line. Find the value of $k$.

OCR MEI AS Paper 1 2024 June Q4

4 marks Challenging +1.2

4 The perpendicular lines AC and BD intersect at E as shown in the diagram. The point E is the midpoint of AC . The angles BAC and BDC are each equal to $\chi ^ { \circ }$. The lengths of AB and CD are 4 cm and 7 cm respectively. \includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-3_606_529_1370_244} Determine the value of $x$.

OCR MEI AS Paper 1 2020 November Q8

7 marks Standard +0.3

8 The point A has coordinates $( - 1 , - 2 )$ and the point B has coordinates (7,4). The perpendicular bisector of $A B$ intersects the line $y + 2 x = k$ at $P$. Determine the coordinates of P in terms of $k$.

Edexcel C1 Q9

12 marks Standard +0.3

9. The points $A ( - 1 , - 2 ) , B ( 7,2 )$ and $C ( k , 4 )$, where $k$ is a constant, are the vertices of $\triangle A B C$. Angle $A B C$ is a right angle.

Find the gradient of $A B$.
Calculate the value of $k$.
Show that the length of $A B$ may be written in the form $p \sqrt { 5 }$, where $p$ is an integer to be found.
Find the exact value of the area of $\triangle A B C$.
Find an equation for the straight line $l$ passing through $B$ and $C$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C1 Q8

12 marks Standard +0.3

8. The points $A ( - 1 , - 2 ) , B ( 7,2 )$ and $C ( k , 4 )$, where $k$ is a constant, are the vertices of $\triangle A B C$. Angle $A B C$ is a right angle.

Find the gradient of $A B$.
Calculate the value of $k$.
Show that the length of $A B$ may be written in the form $p \sqrt { 5 }$, where $p$ is an integer to be found.
Find the exact value of the area of $\triangle A B C$.
Find an equation for the straight line $l$ passing through $B$ and $C$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

AQA C1 2007 June Q1

8 marks Moderate -0.8

1 The points $A$ and $B$ have coordinates $( 6 , - 1 )$ and $( 2,5 )$ respectively.

1. Show that the gradient of $A B$ is $- \frac { 3 } { 2 }$.
2. Hence find an equation of the line $A B$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
1. Find an equation of the line which passes through $B$ and which is perpendicular to the line $A B$.
2. The point $C$ has coordinates ( $k , 7$ ) and angle $A B C$ is a right angle. Find the value of the constant $k$.

Edexcel AEA 2024 June Q2

6 marks Challenging +1.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_904_826_255_623} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the curve defined by the equation $$y ^ { 2 } + 3 y - 6 \sin y = 4 - x ^ { 2 }$$ The point $P ( x , y )$ lies on the curve.
The distance from the origin,$O$ ,to $P$ is $D$ .

Write down an equation for $D ^ { 2 }$ in terms of $y$ only.
Hence determine the minimum value of $D$ giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_2266_53_312_1977}

CAIE P1 2011 June Q7

7 marks Moderate -0.3

The line $L_1$ passes through the points $A(2, 5)$ and $B(10, 9)$. The line $L_2$ is parallel to $L_1$ and passes through the origin. The point $C$ lies on $L_2$ such that $AC$ is perpendicular to $L_2$. Find

the coordinates of $C$, [5]
the distance $AC$. [2]

CAIE P1 2017 June Q8

8 marks Moderate -0.3

$A(-1, 1)$ and $P(a, b)$ are two points, where $a$ and $b$ are constants. The gradient of $AP$ is 2.

Find an expression for $b$ in terms of $a$. [2]
$B(10, -1)$ is a third point such that $AP = AB$. Calculate the coordinates of the possible positions of $P$. [6]

CAIE P1 2016 November Q6

7 marks Moderate -0.3

Three points, $A$, $B$ and $C$, are such that $B$ is the mid-point of $AC$. The coordinates of $A$ are $(2, m)$ and the coordinates of $B$ are $(n, -6)$, where $m$ and $n$ are constants.

Find the coordinates of $C$ in terms of $m$ and $n$. [2]

The line $y = x + 1$ passes through $C$ and is perpendicular to $AB$.

Find the values of $m$ and $n$. [5]

CAIE P3 2018 June Q8

7 marks Standard +0.3

Points $A$ and $B$ have coordinates $(h, h)$ and $(4h + 6, 5h)$ respectively. The equation of the perpendicular bisector of $AB$ is $3x + 2y = k$. Find the values of the constants $h$ and $k$. [7]