CIE Coordinate Geometry (-2020)

1. (CIE $0606 / 2018 / \mathrm{w} / 12 / \mathrm{q} 12)$

The line $y=2 x+5$ intersects the curve $y+x y=5$ at the points $A$ and $B$. Find the coordinates of the point where the perpendicular bisector of the line $A B$ intersects the line $y=x$.

2. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 21 / \mathrm{q} 10)$

The line $y=12-2 x$ is a tangent to two curves. Each curve has an equation of the form $y=k+6+k x-x^{2}$, where $k$ is a constant.

(i) Find the two values of $k$.

The line $y=12-2 x$ is a tangent to one curve at the point $A$ and the other curve at the point $B$.

(ii) Find the coordinates of $A$ and of $B$.

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

3. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 22 / \mathrm{q} 10)$

Two lines are tangents to the curve $y=12-4 x-x^{2}$. The equation of each tangent is of the form $y=2 k+1-k x$, where $k$ is a constant.

(i) Find the two possible values of $k$.

(ii) Find the coordinates of the point of intersection of the two tangents.

4. (CIE $0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 2)$

Do not use a calculator in this question.

Find the coordinates of the points of intersection of the curve $y=(2 x+3)^{2}(x-1)$ and the line $y=3(2 x+3)$

5. (CIE $0606 / 2019 / \mathrm{s} / 21 / \mathrm{q} 10)$

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

The points $A$ and $B$ have coordinates $(p, 3)$ and $(1,4)$ respectively and the line $L$ has equation $3 x+y=2$.

(i) Given that the gradient of $A B$ is $\frac{1}{3}$, find the value of $p$. [2]

(ii) Show that $L$ is the perpendicular bisector of $A B$. $[3]$

(iii) Given that $C(q,-10)$ lies on $L$, find the value of $q$. $[1]$

(iv) Find the area of triangle $A B C$.

6. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 21 / \mathrm{q} 6)$

Do not use a calculator in this question.

The curve $x y=11 x+5$ cuts the line $y=x+10$ at the points $A$ and $B$. The mid-point of $A B$ is the point $C$. Show that the point $C$ lies on the line $x+y=11$.

7. $(\mathrm{CIE} 0606 / 2019 / \mathrm{m} / 22 / \mathrm{q} 5)$

Solutions to this question by aecurate drawing will not be aceepted.

The points $A(3,2), B(7,-4), C(2,-3)$ and $D(k, 3)$ are such that $C D$ is perpendicular to $A B$. Find the equation of the perpendicular bisector of $C D$.

8. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 22 / \mathrm{q} 9)$

The diagram shows the points $A(-3,5)$ and $B(5,-1)$. The mid-point of $A B$ is $M$ and the line $P M$ is perpendicular to $A B$. The point $P$ has coordinates $(r, s)$.

(i) Find the equation of the line $P M$ in the form $y=m x+c$, where $m$ and $c$ are exact constants.

(ii) Hence find an expression for $s$ in terms of $r$. $[1]$

(iii) Given that the length of $P M$ is 10 units, find the value of $r$ and of $s$. [5]

9. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 23 / \mathrm{q} 3)$

The points $A, B$ and $C$ have coordinates $(4,7),(-3,9)$ and $(6,4)$ respectively.

(i) Find the equation of the line, $L$, that is parallel to the line $A B$ and passes through $C$. Give your answer in the form $a x+b y=c$, where $a, b$ and $c$ are integers.

(ii) The line $L$ meets the $x$-axis at the point $D$ and the $y$-axis at the point $E$. Find the length of $D E$.

10. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 11 / \mathrm{q} 6)$

The line $y=5 x+6$ meets the curve $x y=8$ at the points $A$ and $B$.

(a) Find the coordinates of $A$ and of $B$.

(b) Find the coordinates of the point where the perpendicular bisector of the line $A B$ meets the line $y=x$

11. $(\mathrm{CIE} 0606 / 2020 / \mathrm{m} / 12 / \mathrm{q} 6)$

Solutions by accurate drawing will not be accepted.

The points $A$ and $B$ have coordinates $(-2,4)$ and $(6,10)$ respectively.

(a) Find the equation of the perpendicular bisector of the line $A B$, giving your answer in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers.

The point $C$ has coordinates $(5, p)$ and lies on the perpendicular bisector of $A B$.

(b) Find the value of $p$.

It is given that the line $A B$ bisects the line $C D$.

(c) Find the coordinates of $D$.

12. (CIE $0606 / 2020 / \mathrm{w} / 22 / \mathrm{q} 3)$

(a) Find the equation of the perpendicular bisector of the line joining the points $(12,1)$ and $(4,3)$, giving your answer in the form $y=m x+c$.

(b) The perpendicular bisector cuts the axes at points $A$ and $B$. Find the length of $A B$.

13. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 22 / \mathrm{q} 5)$

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

The points $A$ and $B$ are $(4,3)$ and $(12,-7)$ respectively.

(a) Find the equation of the line $L$, the perpendicular bisector of the line $A B$.

(b) The line parallel to $A B$ which passes through the point $(5,12)$ intersects $L$ at the point $C$. Find the coordinates of $C$. [4]

14. $($ CIE $0606 / 2020 / \mathrm{m} / 22 / \mathrm{q} 7)$

The diagram shows the graph of $\mathrm{f}(x)=a \cos b x+c$ for $0 \leqslant x \leqslant \frac{8 \pi}{3}$ radians.

(a) Explain why $\mathrm{f}$ is a function. [1]

(b) Write down the range of $f$.

(c) Find the value of each of the constants $a, b$ and $c$. $[1]$ $[4]$

15. (CIE $0606 / 2020 / \mathrm{s} / 23 / \mathrm{q} 1)$

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

Find the equation of the perpendicular bisector of the line joining the points $(4,-7)$ and $(-8,9)$. [4]

1. $\left(\frac{5}{12}, \frac{5}{12}\right)$

2. (i) $k=-10$ or 2

(ii) $(-4,20),(2,8)$

(iii) $y=\frac{1}{2} x+14.5$

3. (i) $k=6$ or 10

(ii) $x=2, y=1$

4. intersection points $\left(-\frac{3}{2}, 0\right),\left(\frac{3}{2}, 18\right),(-2,-3)$

5. (i) $-2$ (ii) Show

(iii) $q=4$ (iv) $22.5$

6. Show

7. $y=-\frac{3}{2}(x-6.5)$

8. (i) $y=\frac{4}{3} x+\frac{2}{3}$

(ii) $s=\frac{4}{3} r+\frac{2}{3}$

(iii) $r=7, s=10$

9. (i) $2 x+7 y=40$ (ii) $20.8$

10. (a) $A=\left(\frac{4}{5}, 10\right) \quad B=(-2,-4)$

(b) $\left(\frac{12}{5}, \frac{12}{5}\right)$

11. (a) $4 x+3 y-29=0$

(b) $p=3$ (c) $(-1,11)$

12. (a) $y=4 x-30$

(b) $A B=30.9$

13. (a) $y+2=\frac{4}{5}(x-8)$

(b) $C=(13,2)$

14. $(0.5,0.25)$ and $(2,1)$

15. $y-1=\frac{3}{4}(x+2)$