坐標幾何

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

In the figure, the straight line $L_1:y=ax+b$ and the straight line $L_2:y=cx+d$ intersect at a point on the positive $x$-axis. Which of the following must be true?

The coordinates of two vertices of a triangle are $(-4,-8)$ and $(6,2)$. If the coordinates of the circumcentre of the triangle are $(k,-4)$, then $k=$

In the figure, the circle touches the positive $x$-axis and the positive $y$-axis. The coordinates of the points $A$ and $B$ are $(21,0)$ and $(0,28)$ respectively. If $AB$ passes through the centre of the circle, find the equation of the circle.

The coordinates of the points $A$ and $B$ are $(-2,a)$ and $(b,7)$ respectively. If the coordinates of the mid-point of $AB$ are $(1,5)$, then $a=$

If the polar coordinates of the point $P$ are $(2,300^\circ)$, then the rectangular coordinates of $P$ are

In the figure, the straight lines $L_1$ and $L_2$ are perpendicular to each other. Find the equation of $L_2$.

In the figure, $L_1$, $L_2$, $L_3$ and $L_4$ are straight lines. If $m_1$, $m_2$, $m_3$ and $m_4$ are the slopes of $L_1$, $L_2$, $L_3$ and $L_4$ respectively, which of the following must be true?

Let $O$ be the origin. If the coordinates of the points $A$ and $B$ are $(48,0)$ and $(24,18)$ respectively, then the $y$-coordinate of the orthocentre of $\triangle ABO$ is

The equation of a circle is $x^2+y^2-4x-8y+11=0$. Which of the following are true? I. The coordinates of the centre of the circle are $(2,4)$. II. The radius of the circle is $3$. III. The origin lies outside the circle.