2019 Nov P2 Q4

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.	$(5)$
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