Further Pure Math (Coordinate Geometry)

 1. $(2011 /$ june $/$ paper01/q8)

The points $A$ and $B$ have coordinates $(1,5)$ and $(9,7)$ respectively.

(a) Find an equation of $A B$, giving your answer in the form $y=a x+b$, where $a$ and $b$ are rational numbers.

The line $l$ is the perpendicular bisector of $A B$.

(b) Find an equation of $l .$ (4)

The point $C$ has coordinates $(3, q)$. Given that $C$ lies on $l$

(c) find the value of $q$.

The line $l$ meets the $x$-axis at the point $D$.

(d) Find the exact area of the kite $A C B D$.


2. $(2012 / \mathrm{jan} /$ paper02 $/ \mathrm{q} 3)$

Find the coordinates of the points of intersection of the curve with equation $y=3+6 x-x^{2}$ and the line with equation $y-x=7$


3. $(2012 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 7)$

The points $A, B$ and $C$ have coordinates $(3,5),(7,8)$ and $(6,1)$ respectively.

(a) Show, by calculation, that $A B$ is perpendicular to $A C$.

(b) Find an equation for $A C$ in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers whose values must be stated.

The point $D$ is on $A C$ produced and $A C: C D=1: 2$

(c) Find the coordinates of $D$ (2)

(d) Calculate the area of triangle $A B D$. (4)


4. (2012/june/paper01/q10)

The point $A$ has coordinates $(-3,4)$ and the point $C$ has coordinates $(5,2) .$ The mid-point of $A C$ is $M$. The line $l$ is the perpendicular bisector of $A C$.

(a) Find an equation of $l$.

(b) Find the exact length of $A C$.

The point $B$ lies on the line $l .$ The area of triangle $A B C$ is $17 \sqrt{2}$

(c) Find the exact length of $B M$.

(d) Find the exact length of $A B$.

(e) Find the coordinates of each of the two possible positions of $B$.  (6)


5. (2013/jan/Paper01/q7)

The point $C$ with coordinates $(2,1)$ is the centre of a circle which passes through the point $A$ with coordinates $(3,3)$

(a) Find the radius of the circle.

The line $A B$ is a diameter of the circle.

(b) Find the coordinates of $B$. (2)

The points $D$ with coordinates $(0,2)$ and $E$ with coordinates $(4,0)$ lie on the circle.

(c) Show that $D E$ is a diameter of the circle.

The point $P$ has coordinates $(x, y)$.

(d) Find an expression, in terms of $x$ and $y$, for the length of $C P$. $(2)$

Given that the point $P$ lies on the circle,

(e) show that $x^{2}+y^{2}-4 x-2 y=0$ (2)


6. $(2013 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 7)$

The line $/$ passes through the points with coordinates $(1,6)$ and $(3,2)$.

(a) Show that an equation of $l$ is $y+2 x=8$

The curve $C$ has equation $x y=8$

(b) Show that $l$ is a tangent to $C$.

Given that $l$ is the tangent to $C$ at the point $A$,

(c) find the coordinates of $A$. (2)

(d) Find an equation, with integer coefficients, of the normal to $C$ at $A$.


7. (2013/june/paper02/q8)

The equation of line $l_{1}$ is $2 x+3 y+6=0$

(a) Find the gradient of $l_{1}$

The line $l_{2}$ is perpendicular to $l_{1}$ and passes through the point $P$ with coordinates $(7,2)$.

(b) Find an equation for $l_{2}$ (3)

The lines $l_{1}$ and $l_{2}$ intersect at the point $Q$.

(c) Find the coordinates of $Q$. (3)

The line $l_{3}$ is parallel to $l_{1}$ and passes through the point $P$.

(d) Find an equation for $l_{3}$

The line $l_{1}$ crosses the $x$-axis at the point $R$.

(e) Show that $P Q=Q R$. (3)

The point $S$ lies on $l_{3}$

The line $P R$ is perpendicular to $Q S$.

(f) Find the exact area of the quadrilateral $P Q R S$. (3)


8. (2014/jan/paper02/q1)

The points $A$ and $B$ have coordinates $(5,9)$ and $(9,3)$ respectively. The line $l$ is the perpendicular bisector of $A B$.

Find an equation for $l$ in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers.


9. (2014/june/paper01/q9)

The points $A$ and $B$ have coordinates $(2,5)$ and $(16,12)$ respectively. The point $D$ with coordinates $(8,8)$ lies on $A B$.

(a) Find, in the form $p: q$, the ratio in which $D$ divides $A B$ internally.

The line $l$ passes through $D$ and is perpendicular to $A B$.

(b) Find an equation of $l$. (4)

The point $E$ with coordinates ( $e, 6)$ lies on $l$.

(c) Find the value of $e$.

The line $E D$ is produced to $F$ so that $E D=D F$.

(d) Find the coordinates of $F$.

(e) Find the area of the kite $A E B F$. (3)


10. (2014/june/paper02/q4)

(a) Find the coordinates of the points where the line with equation $y=4 x-4$ meets the curve with equation $y=x^{2}-3 x+6$

(b) Hence, or otherwise, find the set of values of $x$ for which $x^{2}-3 x+6 \geqslant 4 x-4$


11. (2015/jan/paper01/q10)

The points $A, B$ and $C$ have coordinates $(-2,3),(2,5)$ and $(4,1)$ respectively.

(a) Show, by calculation, that $A B$ is perpendicular to $B C$.

(b) Show that the length of $A B=$ the length of $B C$.

The midpoint of $A C$ is $M$.

(c) Find the coordinates of $M$.

(d) Find the exact length of the radius of the circle which passes through the points $A, B$ and $C$.

The point $P$ lies on $B M$ such that $B P: P M=2: 1$

(e) Find the coordinates of $P$.

The point $Q$ lies on $A P$ produced such that $A P: P Q=2: 1$

(f) Find the coordinates of $Q$.

(g) Show that $Q$ lies on $B C$.


12. (2015/june/paper02/q9)

The points $A$ and $B$ have coordinates $(2,9)$ and $(10,3)$ respectively.

The point $M$ is the midpoint of $A B$.

(a) Find the coordinates of $M$. (2)

(b) Find the length of $A B$

The line $l$ is the perpendicular bisector of $A B$.

(c) Find an equation for $l$ giving your answer in the form $a y=b x+c$, where $a, b$ and $c$ are integers.

The point $D$ lies on $l$ and has coordinates $(d, 2)$.

(d) Find the value of $d$.

The point $E$ lies on $l$ and is such that $D M: M E=1: 2$

(e) Find the coordinates of $E$,

(f) Find the area of the kite $A E B D$.


13. (2016/jan/paper01/q11)

$$\mathrm{f}(x)=4+3 x-x^{2}$$

(a) Write $\mathrm{f}(x)$ in the form $P-Q(x+R)^{2}$, where $P, Q$ and $R$ are rational numbers.

The curve $C$ has equation $y=4+3 x-x^{2}$

(b) Find the coordinates of the maximum point of $C$.(1)

The line $l_{1}$ is a tangent to $C$ at the point where $x=1$

(c) Find an equation for $l_{1}$

Another line $l_{2}$ is perpendicular to $l_{1}$ and is also a tangent to $C$.

The lines $l_{1}$ and $l_{2}$ intersect at the point $A$.

(d) Find the coordinates of $A$.$(5)$

The point $B$ with coordinates $(-3,2)$ lies on $l_{1}$

(e) Find the exact length of $A B$.$(2)$

The point $D$ with coordinates $(8,0)$ lies on $l_{2}$

(f) Find the exact area of triangle $A B D$.(3)


14. (2016/june/paper01/q10)

The points $A$ and $B$ have coordinates $(2,4)$ and $(5,-2)$ respectively.

The point $C$ divides $A B$ in the ratio $1: 2$

(a) Find the coordinates of $C$. (2)

The point $D$ has coordinates $(1,1)$

(b) Show that $D C$ is perpendicular to $A B$. (3)

(c) Find the equation of $D C$ in the form $p y=x+q$

The point $E$ is such that $D C E$ is a straight line and $D C=C E$.

(d) Find the coordinates of $E$. (2)

(e) Calculate the area of quadrilateral $A D B E$.


15. (2017/jan/paper01/q11)

The curve $C$ has equation $y=p x+q x^{2}$ where $p$ and $q$ are integers.

The curve $C$ has a stationary point at $(3,9)$.

(a) (i) Show that $p=6$ and find the value of $q$.

(ii) Determine the nature of the stationary point at $(3,9)$. (7)

The straight line $l$ with equation $y+x-10=0$ intersects $C$ at two points.

(b) Determine the $x$ coordinate of each of these two points of intersection.

The finite region bounded by the curve $C$ and the straight line $l$ is rotated through $360^{\circ}$ about the $x$-axis.

(c) Use algebraic integration to find the volume of the solid formed. Give your answer in terms of $\pi$.


16. (2017/jan/paper02/q9)

The points $P$ and $Q$ have coordinates $(-2,5)$ and $(2,-3)$ respectively.

(a) Find an equation for the line $P Q$. $(2)$

The point $N$ is such that $P N Q$ is a straight line and $P N: N Q=3: 1$

The straight line $l$ passes through $N$ and is perpendicular to $P Q$.

(b) Find

(i) the coordinates of $N$,

(ii) an equation for $l$. (5)

The points $S$ and $T$ lie on $l$ and have coordinates $(3, s)$ and $(t,-2)$ respectively.

(c) Find

(i) the value of $s$,

(ii) the value of $t$. (2)

(d) Find the area of the quadrilateral $P S Q T$.


17. (2017/june/paper01/q8)

The points $A$ and $B$ have coordinates $(1,7)$ and $(13,1)$ respectively.

(a) Find the exact length of $A B$, (2)

The point $C$ divides $A B$ in the ratio $1: 2$

(b) Find the coordinates of $C$. (2)

The line $l$ passes through $C$ and is perpendicular to $A B$.

(c) Find an equation of $l$, giving your answer in the form $y=a x+b$ where $a$ and $b$ are integers.

The point $D$ with coordinates $(9, d)$ lies on $l$.

(d) Find the value of $d$.

The point $E$ is the midpoint of $C D$.

(e) Find the exact value of the area of the quadrilateral $A D B E$. (5)


18. (2018/jan/paper02/q10)

The point $A$ has coordinates $(-6,-4)$ and the point $B$ has coordinates $(4,1)$ The line $l$ passes through the point $A$ and the point $B$.

(a) Find an equation of $l$. (2)

The point $P$ lies on $l$ such that $A P: P B=3: 2$

(b) Find the coordinates of $P$ (2)

The point $Q$ with coordinates $(m, n)$ lies on the line through $P$ that is perpendicular to $l$. Given that $m<0$ and that the length of $P Q$ is $3 \sqrt{5}$

(c) find the coordinates of $Q$. (5)

The point $R$ has coordinates $(-13,0)$

(d) Show that

(i) $A B$ and $R Q$ are equal in length,

(ii) $A B$ and $R Q$ are parallel.

(e) Find the area of the quadrilateral $A B Q R$.


19. (2018/june/paper02/q9)

The points $A, B$ and $C$ have coordinates $(-4,4),(1,6)$ and $(-2,-1)$ respectively.

(a) Show, by calculation, that $A B$ is perpendicular to $A C$.

(b) Find an equation for $B C$ in the form $p x+q y+r=0$, where $p, q$ and $r$ are integers.

The line $l$ is the perpendicular bisector of $A B$.

(c) Find an equation for $l$.

The line $l$ and the line $B C$ intersect at the point $E$.

(d) Find the coordinates of $E$. (2)

(e) Calculate the area of triangle $A E C$. (4)


20. (2019/june/paper02/q8)

The point $A$ has coordinates $(2,6)$, the point $B$ has coordinates $(6,8)$ and the point $C$ has coordinates $(4,2)$

(a) Find the exact length of

(i) $A B$

(ii) $B C$

(iii) $A C$

(b) Find the size of each angle of triangle $A B C$ in degrees.

The points $A, B$ and $C$ lie on a circle with centre $P$.

(c) Find the coordinates of $P$.

(d) Find the exact length of the radius of the circle in the form $\sqrt{a}$, where $a$ is an integer. (2)


21. (2019/juneR/paper01/q11)

The points $A$ and $B$ have coordinates $(-1,3)$ and $(5,6)$ respectively.

(a) Find an equation for the line $A B$. (2)

The point $P$ divides $A B$ in the ratio $2: 1$

(b) Show that the coordinates of $P$ are $(3,5)$

The point $C$ with coordinates $(m, n)$, where $m>0$, is such that $C P$ is perpendicular to the line $A B$.

Given that the radius of the circle which passes through $A, P$ and $C$ is 5

(c) find the value of $m$ and the value of $n$

The point $D$ with coordinates ( $p, q)$ is such that the line $A D$ is perpendicular to the line $A B$ and the line $D C$ is parallel to the line $A B$.

(d) Find the value of $p$ and the value of $q$. (3)

(e) Find the area of trapezium $A B C D$.







Answer


1.(a) $y=\frac{1}{4} x+\frac{19}{4}$ (b) $y=-4 x+26$ (c) $q=14$ (d) $59 \frac{1}{2}$

2.) $(1,8),(4,11)$

3.(a) Show (b) $3 y+4 x-27=0$ (c) $(12,-7)$ (d) $37 \displaystyle\frac{1}{2}$

4.(a) $y=4 x-1$ (b) $A C=\sqrt{68}$ (c) $B M=\sqrt{34}$ (d) $A B=\sqrt{51}$ (e) $(1+\sqrt{2}, 3+4 \sqrt{2}),(1-\sqrt{2}, 3-4 \sqrt{2})$

5.(a) $r=\sqrt{5}$ (b) $B(1,-1)$ (c) $D E=2 \sqrt{5}$ (d) $C P=\sqrt{(x-2)^{2}+(y-1)}$

6(a) show (b) show (c) $(2,4)$ (d) $2 y=x+6$

7.(a) $-\displaystyle\frac{2}{3}$ (b) $2 y=3 x-17$ (c) $Q(3,-4)$ (d) $3 y+2 x=20$ (e) Show (f) 52

8. $2 x-3 y+4=0$

9.(a) $\displaystyle\frac{3}{4}$ (b) $y+2 x=24$ (c) $e=9$ (d) $F(7,10)$ (e) 35

10.(a) $(2,4),(5,16)$ (b) $x \leqslant 2$ or $x \geqslant 5$

11.(a) Show (b) Show (c) $M(1,2)$ (d) $\sqrt{10}$ (e) $\left(\displaystyle\frac{4}{3},3\right)$ (f) (3,3) (g) Show

12.(a)(6,6) (b) 10 (c) $3 y=4 x-6$ (d) $d=3$ (e) (12,14) (f) 75

13.(a) $ \displaystyle\frac{25}{4}-\left(x-\displaystyle\frac{3}{2}\right)^{2}$ (b) $\left(\displaystyle\frac{3}{2}, \displaystyle\frac{25}{4}\right)$ (c) $y=x+5$ (d) $\left(\displaystyle\frac{3}{2}, \displaystyle\frac{13}{2}\right)$ (e) $A B=\displaystyle\frac{9 \sqrt{2}}{2}$ (f) $A=\displaystyle\frac{117}{4}$

14.(a) $C=(3,2)$ (b) Show (c) $2 y=x+1$ (d) $E=(5,3)$ (e) 15

15.(a) (i) Show (ii) $\max$(b) $x=2,5$(c) $V=\displaystyle\frac{333 \pi}{5}$

16. (a) $y=-2 x+1(b)(i)(1,-1)$ (ii) $2 y=x-3$ (c) (i) $s=0$ (ii) $t=-1$ (d) 20

17. $(a) A B=6 \sqrt{5}$ (b) $C=(5,5)$ (c) $y=2 x-5$ (d) $d=13$ (e) 30

18.(a) $y=\displaystyle\frac{1}{2}x-1$ (b) $(0,-1)$ (c) $(-3,5)$ (d) Show

19.(a) Show (b) $7x-3y+11=0$ (c) $4y=-10x+5$ (d) $\left(-\displaystyle\frac{1}{2},\displaystyle\frac{5}{2}\right)$ (e) 7.25

20. $(a)\left(\begin{array}{lll}\text { i }) A B=\sqrt{20} & \text { (ii) } B C=\sqrt{40} & \text { (iii) } A C=\sqrt{20}\end{array}\right.$ (b) $\angle A=90^{\circ}, \angle B=\angle C=45^{\circ}$ (c) $P=(5,5)$ (d) radius $=\sqrt{10}$

21.(a) $2y=x+7$ (b) Show (c) $m=7,n=-3$ (d) $p=3,q=-5$ (e) $50$


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