坐標幾何

The circle $x^2+y^2-2x-7y-8=0$ intersects the $x$-axis at $A$ and $B$. Find the length of $AB$.

The equations of two circles are $x^2+y^2+ax-by=0$ and $x^2+y^2-ax+by=0$. Which of the following must be true? I. The two circles have the same centre. II. The two circles have equal radii. III. The line joining the centres of the two circles passes through the origin.

$A(7,14)$ and $B(1,2)$ are two points. $C$ is a point on $AB$ produced such that $AB:BC=2:1$. Find the coordinates of $C$.

Which of the following lines is perpendicular to the line $\dfrac{x}{2}+\dfrac{y}{3}=1$ ?

In the figure, find the coordinates of the mid-point of $AB$.

The equation of a circle is given by $x^2+y^2-4x+6y-3=0$. Which of the following statements is/are true? I. The centre of the circle is $(-2,3)$. II. The radius of the circle is $4$. III. The origin is inside the circle.

A circle has $(a,0)$ and $(0,b)$ as the end points of a diameter. Which of the following points lie(s) on this circle? I. $(-a,-b)$ II. $(0,0)$ III. $(a,b)$

In the figure, $AEB$ and $ADC$ are straight lines. $ED\parallel BC$ and $ED:BC=2:3$. If the coordinates of $A$ and $B$ are $(4,7)$ and $(0,1)$ respectively, find the coordinates of $E$.

If $a$, $b$ and $c$ are all positive, which of the following may represent the graph of $ax+by+c=0$ ?

The equation of the circle centred at $(a,b)$ and tangential to the $x$-axis is