Coordinate Geometry (CIE) (2015-2018)

$\let\frac=\dfrac$

22  (CIE $2015, \mathrm{~s}$, paper 11, question 5)

The curve $y=xy+x^2-4$ intersect the line $y=3x-1$ at the point $A$ and $B$. Find the equation of the perpendicular of the line $AB$.

23 (CIE $2015, \mathrm{~s}$, paper 22 , question 9)

Solutions by accurate drawing will not be accepted.

The line $y=m x+4$ meets the lines $x=2$ and $x=-1$ at the points $P$ and $Q$ respectively. The point $R$ is such that $O R$ is parallel to the $y$ -axis and the gradient of $R P$ is 1 . The point $P$ has coordinates $(2,10)$.

(i) Find the value of $m$. [2]

(ii) Find the $y$ -coordinate of $Q$. [1]

(iii) Find the coordinates of $R$. $[2]$

(iv) Find the equation of the line through $P$, perpendicular to $P Q$, giving your answer in the form $a x+b y=c$, where $a, b$ and $c$ are integers. [3]

(v) Find the coordinates of the midpoint, $M$, of the line $P Q$. [2]

(vi) Find the area of triangle $Q R M$. $[2]$

24 (CIE 2015, w, paper 11, question 12)

12 The line $2 x-y+1=0$ meets the curve $x^{2}+3 y=19$ at the points $A$ and $B$. The perpendicular bisector of the line $A B$ meets the $x$ -axis at the point $C$. Find the area of the triangle $A B C$. $[9]$

25 (CIE $2015, \mathrm{w}$, paper 13 , question 11)

The line $x-y+2=0$ intersects the curve $2 x^{2}-y^{2}+2 x+1=0 \quad$ at the points $A$ and $B$. The perpendicular bisector of the line $A B$ intersects the curve at the points $C$ and $D$. Find the length of the line $C D$ in the form $a \sqrt{5}$, where $a$ is an integer. $\quad[10]$


26 (CIE 2015, w, paper 21, question 8)

Solutions to this question by accurate drawing will not be accepted.

Two points $A$ and $B$ have coordinates $(-3,2)$ and $(9,8)$ respectively.

(i) Find the coordinates of $C$, the point where the line $A B$ cuts the $y$ -axis. [3]

(ii) Find the coordinates of $D$, the mid-point of $A B$. $[1]$

(iii) Find the equation of the perpendicular bisector of $A B$. [2]

The perpendicular bisector of $A B$ cuts the $y$ -axis at the point $E$.

(iv) Find the coordinates of $E$. $[1]$

(v) Show that the area of triangle $A B E$ is four times the area of triangle $E C D$. $[3]$

27 (CIE 2016, march, paper 22, question 8)

The line $2 y=x+2$ meets the curve $3 x^{2}+x y-y^{2}=12$ at the points $A$ and $B$.

(i) Find the coordinates of the points $A$ and $B$. $[5]$

(ii) Given that the point $C$ has coordinates $(0,6)$, show that the triangle $A B C$ is right-angled. $[2]$

28 (CIE 2016, s, paper 11, question 1) Find the value of $k$ for which the curve $y=2 x^{2}-3 x+k$

(i) passes through the point $(4,-7)$, [1]

(ii) meets the $x$ -axis at one point only. [2]

29 (CIE 2016, s, paper 11, question 8) Solutions to this question by accurate drawing will not be accepted.

Three points have coordinates $A(-8,6), B(4,2)$ and $C(-1,7)$. The line through $C$ perpendicular to $A B$ intersects $A B$ at the point $P$.

(i) Find the equation of the line $A B$. [2]

(ii) Find the equation of the line $C P$. [2]

(iii) Show that $P$ is the midpoint of $A B$. [3]

(iv) Calculate the length of $C P$. [1]

(v) Hence find the area of the triangle $A B C$. [2]

30 (CIE 2016, s, paper 22, question 5) The coordinates of three points are $A(-2,6), B(6,10)$ and $C(p, 0)$.

(i) Find the coordinates of $M$, the mid-point of $A B$. $[2]$

(ii) Given that $C M$ is perpendicular to $A B$, find the value of the constant $p$.

(iii) Find angle $M C B$. $[3]$

31 (CIE 2016, s, paper 22, question 8) Find the coordinates of the points of intersection of the curve $4+\frac{5}{y}+\frac{3}{x}=0$ and the line $y=15 x+10 . \quad[6]$


32 (CIE 2017, march, paper 22, question 8) The points $A(3,7)$ and $B(8,4)$ lie on the line $L$. The line through the point $C(6,-4)$ with gradient $\frac{1}{6}$ meets the line $L$ at the point $D$. Calculate

(i) the coordinates of $D$, $[6]$

(ii) the equation of the line through $D$ perpendicular to the line $3 y-2 x=10$. [2]

33 (CIE 2017, s, paper 13, question 5)


The normal to the curve $y=\sqrt{4 x+9}$, at the point where $x=4$, meets the $x$ - and $y$ -axes at the points $A$ and $B$. Find the coordinates of the mid-point of the line $A B .$

34 (CIE 2017, s, paper 21, question 9)

The curve $3 x^{2}+x y-y^{2}+4 y-3=0$ and the line $y=2(1-x)$ intersect at the points $A$ and $B$.

(i) Find the coordinates of $A$ and of $B$. [5]

(ii) Find the equation of the perpendicular bisector of the line $A B$, giving your answer in the form $a x+b y=c$, where $a, b$ and $c$ are integers.

35 (CIE $2017, \mathrm{~s}$, paper 22 , question 8)

Solutions to this question by accurate drawing will not be accepted.

The points $A$ and $B$ are $(-8,8)$ and $(4,0)$ respectively.

(i) Find the equation of the line $A B$. $[2]$

(ii) Calculate the length of $A B$. $[2]$

The point $C$ is $(0,7)$ and $D$ is the mid-point of $A B$.

(iii) Show that angle $A D C$ is a right angle. [3]

The point $E$ is such that $\overrightarrow{A E}=\left(\begin{array}{r}4 \\ -7\end{array}\right)$.

(iv) Write down the position vector of the point $E$.

(v) Show that $A C B E$ is a parallelogram.

36 ( $\mathrm{CIE} 2017, \mathrm{w}$, paper 13 , question 12)

The line $y=2 x+1$ intersects the curve $x y=14-2 y$ at the points $P$ and $Q$. The midpoint of the line $P Q$ is the point $M$.

(i) Show that the point $\left(-10, \frac{23}{8}\right)$ lies on the perpendicular bisector of $P Q$. [9]

The line $P Q$ intersects the $y$ -axis at the point $R$. The perpendicular bisector of $P Q$ intersects the $y$ -axis at the point $S$.

(ii) Find the area of the triangle $R S M$. [3]


37 (CIE 2018, march, paper 22, question 9)

Solutions to this question by accurate drawing will not be accepted.

$P$ is the point $(8,2)$ and $Q$ is the point $(11,6)$.

(i) Find the equation of the line $L$ which passes through $P$ and is perpendicular to the line $P Q . \quad[3]$

The point $R$ lies on $L$ such that the area of triangle $P Q R$ is $12.5$ units $^{2}$.

(ii) Showing all your working, find the coordinates of each of the two possible positions of point $R$.

38 (CIE 2018, s, paper 11, question 2)


Find the equation of the perpendicular bisector of the line joining the points $(1,3)$ and $(4,-5)$. Give your answer in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers.

39 (CIE 2018, s, paper 22, question 4) Find the coordinates of the points where the line $2 y-3 x=6$ intersects the curve $\frac{x^{2}}{4}+\frac{y^{2}}{9}=5 .$ [5]


Answers 

 22. $3 y+x-2=0$

23. $m=3,1,(-1,7), x+3 y=32$

$(\mathrm{v})(1 / 2,11 / 2)$ (vi) $4.5$

24. 125

25. $8 \sqrt{5}$

26. (i) $y=.5 x+3.5, y=3.5$

(ii) $(3,5)$ (iii) $y=-2 x+11$

(iv) $(0,11)(\mathrm{v}) 11.25$

27. (i) $x=-2, y=0: x=2, y=$

28. (i) $-27$ (ii) $9 / 8$

29. (i) $3 y+x=10$

(ii) $y=3 x+10$

(iv) $\sqrt{10}$ (v) 20

30. (i) $(2,8)$ (ii)6

(iii) $26.56,0.4636$

31. $(-.5,2.5),(-1,-5)$

32. (i) $(18,-2)$

(ii) $y+2=-1.5(x-18)$

33. $(3,15 / 2)$

34. (i) $(-1 / 3,8 / 3),(1,0)$

(ii) $6 y-3 x=7$

35. (i) $3 y=-2 x+8$

(ii) $4 \sqrt{13}$

(iv) $-4 \mathrm{i}+\mathrm{j}$

36. (ii) $1.95$

37. $y=-3 / 4 x+8$

$(4,5),(12,-1)$

38. $6 x-16 y-31=0$

39. $(2,6),(-4,-3)$

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