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Analytical-geometry

Written on
November 2020 - Paper 2

Question 3

$\Delta TSK$ is drawn. The equation of $ST$ is $y=\frac{\displaystyle 1}{\displaystyle 2}x+6$ and $ST$ cuts the $x$ -axis at $M$ . $W(-4\,;\,4)$ lies on $ST$ and $R$ lies on $SK$ such that $WR$ is parallel to the $y$ -axis. $WK$ cuts the $x$ -axis at $V$ and the $y$ -axis at $P(0\,;\,-4)$ . $KS$ produced cuts the $x$ -axis at $N$ . $T\hat SK=\theta \,$ .

$3.1$	Calculate the gradient of $WP$ .	$(2)$
$3.2$	Show that $WP\perp ST$ .	$(2)$
$3.3$	If the equation of $SK$ is given as $5y+2x+60=0$ , calculate the coordinates of $S$ .	$(4)$
$3.4$	Calculate the length of $WR$ .	$(3)$
$3.5$	Calculate the size of $\theta$ .	$(5)$
$3.6$	Let $L$ be a point in the third quadrant such that $SWRL$ , in that order, forms a parallelogram. Calculate the area of $SWRL$ .	$(4)$
		$\textbf{[21]}$

Written on
November 2019 - Paper 2

Question 3

In the diagram, $P$ , $R(3\,;\,5)$ , $S(-3\,;\,-7)$ and $T(-5\,;\,k)$ are vertices of a trapezium $PRST$ and $PT \parallel RS$ . $RS$ and $PR$ cut the $y$ -axis at $D$ and $C(0\,;\,5)$ respectively. $PT$ and $RS$ cut the $x$ -axis at $E$ and $F$ respectively. $P\hat EF=\theta$ .

$3.1$	Write down the equation of $PR$ .	$(1)$
$3.2$	Calculate the:
$\quad 3.2.1$	Gradient of $RS$	$(2)$
$\quad 3.2.2$	Size of $\theta$	$(3)$
$\quad 3.2.3$	Coordinates of $D$	$(3)$
$3.3$	If it is given that $TS=2\sqrt{5}$ , calculate the value of $k$ .	$(4)$
$3.4$	Parallelogram $TDNS$ , with $N$ in the $4^{th}$ quadrant, is drawn. Calculate the coordinates of $N$ .	$(3)$
$3.5$	$\Delta PRD$ is reflected about the $y$ -axis to form $\Delta P^\prime R^\prime D^\prime$ . Calculate the size of $R\hat DR^\prime$ .	$(3)$
		$\textbf{[19]}$

Written on
November 2020 - Paper 2

Question 4

$M(-3\,;\,4)$ is the centre of the large circle and a point on the small circle having centre $O(0\,;\,0)$ . From $N(-11\,;\,p)$ , a tangent is drawn to touch the large circle at $T$ with $NT$ is parallel to the $y$ -axis. $NM$ is a tangent to the smaller circle at $M$ with $MOS$ a diameter.

$4.1$	Determine the equation of the small circle.	$(2)$
$4.2$	Determine the equation of the circle centred at $M$ in the form $(x-a)^2+(y-b)^2=r^2$	$(3)$
$4.3$	Determine the equation of $NM$ in the form $y=mx+c$	$(4)$
$4.4$	Calculate the length of $SN$ .	$(5)$
$4.5$	If another circle with centre $B(-2\,;\,5)$ and radius $k$ touches the circle centred at $M$ , determine the value(s) of $k$ , correct to ONE decimal place.	$(5)$
		$\textbf{[19]}$

Written on
November 2019 - Paper 2

Question 4

In the diagram, a circle having centre $M$ touches the $x$ -axis at $A(-1\,;\,0)$ and the $y$ -axis at $B(0\,;\,1)$ . A smaller circle, centred at $N(-\frac{\displaystyle 1}{\displaystyle 2}\,;\,\frac{\displaystyle 3}{\displaystyle 2})$ , passes through $M$ and cuts the larger circle at $B$ and $C$ . $BNC$ is a diameter of the smaller circle. A tangent drawn to the smaller circle at $C$ , cuts the $x$ -axis at $D$ .

$4.1$	Determine the equation of the circle centred at $M$ in the form $(x-a)^2+(y-b)^2=r^2$	$(3)$
$4.2$	Calculate the coordinates of $C$ .	$(2)$
$4.3$	Show that the equation of the tangent $CD$ is $y-x=3$ .	$(4)$
$4.4$	Determine the values of $t$ for which the line $y=x+t$ will NOT touch or cut the smaller circle.	$(3)$
$4.5$	The smaller circle centred at $N$ is transformed such that point $C$ is translated along the tangent to $D$ . Calculate the coordinates of $E$ , the new centre of the smaller circle.	$(3)$
$4.6$	If it is given that the area of quadrilateral $OBCD$ is $2a^2$ square units and $a>0$ , show that $a=\frac{\displaystyle \sqrt{7}}{\displaystyle 2}$ units.	$(5)$
		$\textbf{[20]}$

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