坐標幾何

$PQRS$ is a parallelogram with vertices $P=(0,0)$, $Q=(a,b)$ and $S=(-b,a)$. Find $R$.

$ABCD$ is a line segment. $AB:BC:CD=3:2:1$. If $A=(4,5)$, $D=(10,11)$, find $C$.

If the line $y=mx+b$ and $\dfrac{x}{a}+\dfrac{y}{b}=1$ are perpendicular, find $m$.

In the figure, the slopes of the straight lines $l_1$, $l_2$, $l_3$ and $l_4$ are $m_1$, $m_2$, $m_3$, $m_4$ respectively. Which of the following is true?

In the figure, a circle cuts the $x$-axis at two points 6 units apart. If the circle has centre $(4,5)$, then its equation is

Which of the following systems of inequalities is represented by the shaded region in the figure?

The equation of the straight line perpendicular to $2x+y-3=0$ and passing through $(1,-1)$ is

In the figure, the line $ax-2y+5=0$ passes through the point $(3,4)$. What is the area of the shaded part?

In the figure, $C$ is the centre of the circle $x^2+y^2-8x-7y+12=0$. If the circle cuts the $x$-axis at $A$ and $B$, find the area of $\triangle CAB$.

In the figure, $C$ is the centre of the circle $x^2+y^2-6x-8y+21=0$. $OA$ and $OB$ are tangents. If $\angle AOB=2\theta$, find $\sin\theta$.