Coordinate Geometry: Mathematics 9709 (Cambridge International AS & A Level)

Points $C(-4, 1)$ and $D(2, -7)$ lie on a coordinate grid.

What is the gradient of the line $CD$?

Points $P(1, 2)$ and $Q(9, 8)$ lie on a coordinate grid.

(a) Find the equation of the line $PQ$, giving your answer in the form $y = mx + c$. [3]

(b) Find the equation of the perpendicular bisector of $PQ$, giving your answer in the form $ax + by = k$, where $a$, $b$ and $k$ are integers. [3]

(c) Find the coordinates of the point at which the perpendicular bisector found in part (b) crosses the $x$-axis. [2]

A circle has equation $x^2 + y^2 - 4x + 8y + 11 = 0$.

(a) Find the coordinates of the centre of the circle, and find the radius of the circle. [3]

(b) The point $M(5, -4)$ is given. Show that $M$ lies on the circle. [2]

(c) Find the equation of the tangent to the circle at the point $M$. [3]

Line $l_1$ has equation $2x + 3y = 12$.

Which of the following lines is perpendicular to $l_1$?

A circle has equation $(x - 1)^2 + (y - 2)^2 = 25$. A line has equation $3x + y = 20$.

(a) Find the coordinates of the two points at which the line intersects the circle. [6]

(b) Find the exact length of the chord joining these two points of intersection. [2]

Points $E(1, 3)$ and $F(7, 11)$ lie on a coordinate grid.

What is the distance $EF$?

Points $A(-2, 5)$ and $B(4, -1)$ lie on line $l_1$.

(a) Find the equation of $l_1$, giving your answer in the form $y = mx + c$. [3]

(b) Line $l_2$ is parallel to $l_1$ and passes through the point $C(5, 4)$. Find the equation of $l_2$, giving your answer in the form $y = mx + c$. [2]

(c) Line $l_3$ has equation $y = 2x - 3$. Find the coordinates of the point at which $l_1$ and $l_3$ intersect. [3]

A circle has centre $(3, -2)$ and passes through the point $(7, 1)$.

(a) Find the equation of the circle, giving your answer in the form $(x - a)^2 + (y - b)^2 = r^2$. [3]

(b) Determine whether the point $D(-1, 2)$ lies inside, on, or outside the circle. [3]

(c) The point $(7, 1)$ lies on the circle. Find the coordinates of the point that is diametrically opposite $(7, 1)$ on this circle. [2]

Points $G(-3, 4)$ and $H(5, -2)$ are the endpoints of a diameter of a circle.

(a) Find the equation of the circle, giving your answer in the form $(x - a)^2 + (y - b)^2 = r^2$. [4]

(b) The point $K(4, 5)$ lies on the circle. Using the fact that the angle in a semicircle is $90°$, show that $GK$ is perpendicular to $KH$. [3]

(c) Determine whether the point $L(6, 3)$ also lies on this circle. [2]

A circle has equation $x^2 + y^2 = 8$. A line has equation $y = x + k$, where $k$ is a constant.

(a) Show that substituting $y = x + k$ into the equation of the circle gives $2x^2 + 2kx + (k^2 - 8) = 0$. [2]

(b) Given that the line is a tangent to the circle, use the discriminant of the quadratic in part (a) to find the two possible values of $k$. [4]

(c) For the positive value of $k$ found in part (b), find the coordinates of the point where the line touches the circle. [2]