CIE Coordinate Geometry (-2018)

$\def\D{\displaystyle}$

1 (CIE 2012, s, paper 12, question 11)
The point $\D P$ lies on the line joining $\D A(-1, -5)$ and $\D B(11, 13)$ such that $\D AP = \frac{1}{3} AB.$
(i) Find the equation of the line perpendicular to $\D AB$ and passing through $\D P.$ [5]
The line perpendicular to $\D AB$ passing through $\D P$ and the line parallel to the x-axis passing through $\D B$ intersect at the point $\D Q.$
(ii) Find the coordinates of the point $\D Q.$ [2]
(iii) Find the area of the triangle $\D PBQ.$ [2]

2 (CIE 2012, s, paper 21, question 10)
Solutions to this question by accurate drawing will not be accepted.


The diagram shows a trapezium $\D ABCD$ with vertices $\D A(11, 4), B(7, 7), C(-3, 2)$ and $\D D.$ The side $\D AD$ is parallel to $\D BC$ and the side $\D CD$ is perpendicular to $\D BC.$ Find the area of the trapezium $\D ABCD.$ [9]

3 (CIE 2012, w, paper 11, question 8)
The points $\D A(-3, 6), B(5, 2)$ and $\D C$ lie on a straight line such that $\D B$ is the mid-point of $\D AC.$
(i) Find the coordinates of $\D C.$ [2]
The point $\D D$ lies on the y-axis and the line $\D CD$ is perpendicular to $\D AC.$
(ii) Find the area of the triangle $\D ACD.$ [5]

4 (CIE 2012, w, paper 12, question 5)
The line $\D x - 2y = 6$ intersects the curve $\D x^2 + xy + 10y + 4y^2 = 156$ at the points $\D A$ and $\D B.$ Find the length of $\D AB.$ [7]

5 (CIE 2012, w, paper 12, question 7)
Solutions to this question by accurate drawing will not be accepted.


The vertices of the trapezium $\D ABCD$ are the points $\D A(-5, 4), B(8, 4), C(6, 8)$ and $\D D.$ The line $\D AB$ is parallel to the line $\D DC.$ The lines $\D AD$ and $\D BC$ are extended to meet at $\D E$ and angle $\D AEB = 90^{\circ}.$
(i) Find the coordinates of $\D D$ and of $\D E.$ [6]
(ii) Find the area of the trapezium $\D ABCD.$ [2]

6 (CIE 2012, w, paper 22, question 12either)
The point $\D A(0, 10)$ lies on the curve for which $\D \frac{dy}{dx}=e^{-\frac{\pi}{4}}.$  The point $\D B,$ with x-coordinate $\D -4,$ also lies on the curve.
(i) Find, in terms of $\D e,$ the y-coordinate of $\D B.$ [5]
The tangents to the curve at the points $\D A$ and $\D B$ intersect at the point $\D C.$
(ii) Find, in terms of $\D e,$ the x-coordinate of the point $\D C.$ [5]

7 (CIE 2012, w, paper 23, question 8)
Solutions to this question by accurate drawing will not be accepted.

The points $\D A (4, 5), B(-2, 3), C(1, 9)$ and $\D D$ are the vertices of a trapezium in which $\D BC$ is parallel to $\D AD$ and angle $\D BCD$ is $\D 90^{\circ}.$ Find the area of the trapezium. [8]

8 (CIE 2013, s, paper 12, question 5)
The line $\D 3x + 4y = 15$ cuts the curve $\D 2xy = 9$ at the points $\D A$ and $\D B.$ Find the length of the line $\D AB.$ [6]

9 (CIE 2013, s, paper 21, question 8)
The line $\D y = 2x - 8$ cuts the curve $\D 2x^2 +y^2- 5xy+ 32= 0$  at the points $\D A$ and $\D B.$ Find the length of the line $\D AB.$ [7]

10 (CIE 2013, s, paper 22, question 8)
Solutions to this question by accurate drawing will not be accepted.

The points $\D A(- 6, 2), B(2, 6)$ and $\D C$ are the vertices of a triangle.
(i) Find the equation of the line $\D AB$ in the form $\D y = mx + c.$  [2]
(ii) Given that angle $\D ABC = 90^{\circ},$ find the equation of $\D BC.$ [2]
(iii) Given that the length of $\D AC$ is 10 units, find the coordinates of each of the two possible positions of point $\D C.$ [4]

11 (CIE 2013, s, paper 22, question 9)
(a) The graph of $\D y = k(3^x) + c$ passes through the points $\D (0, 14)$ and $\D (- 2, 6).$ Find the value of $\D k$
and of $\D c.$ [3]
(b) The variables $\D x$ and $\D y$ are connected by the equation $\D y = e^x + 25 - 24e^{-x}.$
(i) Find the value of $\D y$ when $\D x = 4.$ [1]
(ii) Find the value of $\D e^x$ when $\D y = 20$ and hence find the corresponding value of $\D x.$ [4]

12 (CIE 2013, w, paper 11, question 10)
Solutions to this question by accurate drawing will not be accepted.
The points $\D A(-3, 2)$ and $\D B(1, 4)$ are vertices of an isosceles triangle $\D ABC,$ where angle $\D B = 90^{\circ}.$
(i) Find the length of the line $\D AB.$ [1]
(ii) Find the equation of the line $\D BC.$ [3]
(iii) Find the coordinates of each of the two possible positions of $\D C.$ [6]

13 (CIE 2013, w, paper 23, question 7)
The line $\D 4x + y = 16$ intersects the curve $\D \frac{4}{x}-\frac{8}{y}=1$ at the points $\D A$ and $\D B.$ The x-coordinate of $\D A$ is less than the x-coordinate of $\D B.$ Given that the point $\D C$ lies on the line $\D AB$ such that
$\D AC : CB = 1 : 2,$ find the coordinates of $\D C.$ [8]

14 (CIE 2013, w, paper 23, question 8)
Solutions to this question by accurate drawing will not be accepted.

The diagram shows a quadrilateral $\D ABCD,$ with vertices $\D A(- 4, 6), B(6, - 4), C(10, 4)$ and $\D D.$ The angle $\D ADC = 90^{\circ}.$ The lines $\D BC$ and $\D AD$ are extended to intersect at the point $\D X.$
(i) Given that C is the midpoint of BX, find the coordinates of D. [7]
(ii) Hence calculate the area of the quadrilateral $\D ABCD.$ [2]

15 (CIE 2014, s, paper 21, question 9)
Solutions to this question by accurate drawing will not be accepted.

The points $\D A(p,1), B(1, 6), C(4, q) and D(5, 4),$ where $\D p$ and $\D q$ are constants, are the vertices of a kite $\D ABCD.$ The diagonals of the kite, $\D AC$ and $\D BD,$ intersect at the point $\D E.$ The line $\D AC$ is the perpendicular bisector of $\D BD.$ Find
(i) the coordinates of $\D E,$ [2]
(ii) the equation of the diagonal $\D AC,$ [3]
(iii) the area of the kite ABCD. [3]


16 (CIE 2014, s, paper 22, question 8)
The line $\D y = x - 5$ meets the curve $\D x^2+ y^2+ 2x- 35= 0$ at the points $\D A$ and $\D B.$ Find the exact length of $\D AB.$ [6]

17 (CIE 2014, s, paper 23, question 6)
Find the coordinates of the points of intersection of the curve $\D \frac{8}{x}-\frac{10}{y}=1$ and the line $\D x + y = 9.$ [6]

18 (CIE 2014, s, paper 23, question 9)
Solutions to this question by accurate drawing will not be accepted.

The points $\D A(2,11), B(-2, 3)$ and $\D C(2,-1)$ are the vertices of a triangle.
(i) Find the equation of the perpendicular bisector of $\D AB.$ [4]
The line through $\D A$ parallel to $\D BC$ intersects the perpendicular bisector of $\D AB$ at the point $\D D.$
(ii) Find the area of the quadrilateral $\D ABCD.$ [6]

19 (CIE 2014, w, paper 11, question 5)
(i) Find the equation of the tangent to the curve $\D y= x^3- \ln x$  at the point on the curve
where $\D x = 1.$ [4]
(ii) Show that this tangent bisects the line joining the points $\D (-2,16)$ and $\D (12, 2).$ [2]

20 (CIE 2014, w, paper 11, question 8)
The point $\D P$ lies on the line joining $\D A(- 2, 3)$ and $\D B(10, 19)$ such that $\D AP:PB = 1:3.$
(i) Show that the x-coordinate of $\D P$ is 1 and find the y-coordinate of $\D P.$ [2]
(ii) Find the equation of the line through $\D P$ which is perpendicular to $\D AB.$ [3]
The line through $\D P$ which is perpendicular to $\D AB$ meets the y-axis at the point $\D Q.$
(iii) Find the area of the triangle $\D AQB.$ [3]

21 (CIE 2014, w, paper 21, question 6)
(i) Calculate the coordinates of the points where the line $\D y = x + 2$ cuts the curve $\D x^2+y^2=10.$ [4]
(ii) Find the exact values of $\D m$ for which the line $\D y = mx + 5$ is a tangent to the curve $\D x^2+y^2=10.$ [4]

$\let\frac=\dfrac$

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 

Answer

1. (i)$\D 2x + 3y = 9$
(ii) $\D Q(-15; 13)$
(iii) $\D 156$
2. $\D 55$
3. (i) $\D C(13;-2)$
(ii) $\D 260$
4. $\D 8\sqrt{5}$
5. (i) $\D D(3, 8);E(5.4, 9.2)$
(ii) $\D 32$
6. (i) $\D 14 - 4e$
(ii) $\D x = \frac{4}{1-e}$
7. $\D 20$
8. $\D 1.25$
9. $\D 10\sqrt{5}$
10. (i) $\D y = 0.5x + 5$
(ii) $\D y - 6 = -2(x - 2)$
(iii) $\D (0,10),(4,2)$
11. (a) $\D k = 9; c = 5$
(b)(i) $\D 79.2$
(ii) $\D x = \ln 3$
12. (i) $\D \sqrt{20}$
(ii) $\D y = -2x + 6$
(iii) $\D x = 3; y = 0; x = -1; y = 8$
13. $\D (4, 0)$
14. (i) $\D D(8, 10)$
(ii) $\D 100$
15. $\D (3,5),y = 2x - 1,15$
16. $\D 6\sqrt{2}$
17. $\D x = 3; y = 6; x = 24; y = -15$
18. (i) $\D y = -0.5x + 7$
(ii) $\D 84$
19. (i) $\D y = 2x - 1$
20. (i) $\D y = 7$
(ii) $\D 3x + 4y = 31$
(iii) $\D 12.5$
21. (i) $\D (1, 3); (-3,-1)$
(ii) $\D m = \pm \sqrt{1.5}$

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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