Edexcel A-Level Maths: Coordinate Geometry

A circle $C$C has equation $x^2 + y^2 - 8x + 6y = 0.$x2+y2−8x+6y=0.

(a) By completing the square, find the coordinates of the centre of $C$C and the length of its radius. (3 marks)

(b) The line $l$l has equation $y = x - 6$y=x−6. Show that $l$l meets $C$C at two points, and find the coordinates of these two points. (4 marks)

(c) One of the points of intersection found in part (b) is $P(7, 1)$P(7,1). Find an equation of the tangent to $C$C at $P$P, giving your answer in the form $ax + by + c = 0$ax+by+c=0 where $a$a, $b$b and $c$c are integers. (5 marks)

The points $A(-2, 1)$A(−2,1) and $B(6, 5)$B(6,5) lie on a coordinate grid.

(a) Find an equation of the line through $A$A and $B$B, giving your answer in the form $y = mx + c$y=mx+c. (2 marks)

(b) Find an equation of the perpendicular bisector of $AB$AB, giving your answer in the form $y = mx + c$y=mx+c. (4 marks)

(c) The point $C$C has coordinates $(4, -1)$(4,−1). Show that the triangle $ABC$ABC is right-angled at $C$C, and hence find the exact area of triangle $ABC$ABC. (4 marks)

A circle has equation $(x - 2)^2 + (y - 1)^2 = 5$(x−2)2+(y−1)2=5. The line $l$l has equation $y = -2x + 10$y=−2x+10.

(a) Show algebraically that $l$l is a tangent to the circle. (6 marks)

(b) Hence state the coordinates of the point at which $l$l touches the circle. (2 marks)

The line $l_1$l1​ has equation $4x + 3y = 24$4x+3y=24. The line $l_2$l2​ is perpendicular to $l_1$l1​ and passes through the point $P(8, 3)$P(8,3).

(a) Find an equation for $l_2$l2​, giving your answer in the form $y = mx + c$y=mx+c. (4 marks)

(b) Find the coordinates of the point where $l_2$l2​ crosses the $x$x-axis. (2 marks)

A curve is defined by the parametric equations $x = 3t + 1, \qquad y = \frac{2}{t}, \qquad t \in \mathbb{R}, \; t \neq 0.$x=3t+1,y=t2​,t∈R,t=0.

Find a Cartesian equation of the curve in the form $y = \mathrm{f}(x)$y=f(x), and state the value of $x$x that must be excluded from the domain. (5 marks)

The points $A(-3, -4)$A(−3,−4), $B(1, 2)$B(1,2) and $C(5, 8)$C(5,8) are plotted on a coordinate grid.

Show that $A$A, $B$B and $C$C are collinear, and find an equation of the line on which they lie, giving your answer in the form $ax + by + c = 0$ax+by+c=0 where $a$a, $b$b and $c$c are integers. (4 marks)