坐標幾何

In the figure, the equation of the straight line $L$ is

In the figure, the area of $\triangle ABC$ is

In the figure, $x=0$, $y-a=0$ and $y+a=0$ are tangents to the circle. The equation of the circle is

The equation of a circle is given by $(x-a)^2+(y+b)^2=a^2+b^2$, where $a>0$ and $b>0$. Which of the following must be true? I. The centre of the circle is $(a,-b)$. II. The circle passes through the origin. III. The circle cuts the $x$-axis at two distinct points.

If the straight lines $2x-3y+1=0$ and $5x+ky-1=0$ are perpendicular to each other, find $k$.

If the straight lines $x-2y+5=0$ and $ax-y+1=0$ intersect at $(1,b)$, find $a$ and $b$.

In the figure, $A$, $B$ and $C$ are points on a rectangular coordinate plane. $AC$ and $BC$ are parallel to the $x$-axis and $y$-axis respectively. If the coordinates of $C$ are $(2,1)$ and the equation of the straight line $AB$ is $y=2x+3$, find the distance between $A$ and $B$.

Consider the circle $x^2+y^2-8x-6y+21=0$. Find the equation of the chord whose mid-point is $(5,2)$.

In the figure, the inscribed circle of $\triangle OPQ$ touches $PQ$ at $R$. Find the coordinates of $R$.

Consider the three straight lines $L_1:\ 6x+4y-3=0$, $L_2:\ y=-\dfrac{3}{2}x+4$ and $L_3:\ 6x-4y+3=0$. Which of the following is/are true? I. $L_1\parallel L_2$ II. $L_2\parallel L_3$ III. $L_1\perp L_3$