AQA A-Level Maths: Coordinate Geometry and Parametric Equations

A curve $C$C is defined by the parametric equations $x = 3t^2, \qquad y = 2t^3, \qquad t \in \mathbb{R}.$x=3t2,y=2t3,t∈R.

(a) Find $\dfrac{dy}{dx}$dxdy​ in terms of $t$t, and hence find an equation of the tangent to $C$C at the point where $t = 2$t=2, giving your answer in the form $y = mx + c$y=mx+c. (4 marks)

(b) Show that a Cartesian equation of $C$C may be written as $27y^2 = 4x^3$27y2=4x3. (4 marks)

(c) The curve $C$C meets the line $y = x$y=x at the origin and at one other point $Q$Q. Find the coordinates of $Q$Q. (4 marks)

The points $A(-1, 7)$A(−1,7), $B(-3, 3)$B(−3,3) and $C(5, -1)$C(5,−1) lie on a circle.

(a) Find an equation of the perpendicular bisector of $AB$AB, giving your answer in the form $y = mx + c$y=mx+c. (3 marks)

(b) Find an equation of the perpendicular bisector of $BC$BC, and hence determine the coordinates of the centre of the circle and the length of its radius. Write down an equation of the circle. (4 marks)

(c) Find the exact area of triangle $ABC$ABC. (3 marks)

A curve has equation $x^2 + y^2 - xy = 12.$x2+y2−xy=12.

(a) Use implicit differentiation to find $\dfrac{dy}{dx}$dxdy​ in terms of $x$x and $y$y, and hence find the gradient of the curve at the point $(2, -2)$(2,−2). (5 marks)

(b) Find the coordinates of the two points on the curve at which the tangent is horizontal. (3 marks)

A curve $C$C is given parametrically by $x = 2t^2, \qquad y = 3 - t, \qquad 0 \leq t \leq 3.$x=2t2,y=3−t,0≤t≤3.

The region $R$R is bounded by $C$C, the $x$x-axis and the $y$y-axis. Using the result $\displaystyle A = \int y\,\frac{dx}{dt}\,dt$A=∫ydtdx​dt, find the exact area of $R$R. (6 marks)

The points $A(1, 1)$A(1,1), $B(5, 2)$B(5,2), $C(7, 6)$C(7,6) and $D(3, 5)$D(3,5) are the vertices of a quadrilateral.

(a) Show that $ABCD$ABCD is a parallelogram, and determine whether or not it is a rectangle. (3 marks)

(b) Find the exact area of the parallelogram $ABCD$ABCD. (2 marks)

The points $A(0, 0)$A(0,0) and $B(6, 0)$B(6,0) are fixed. A point $P(x, y)$P(x,y) moves so that its distance from $A$A is always twice its distance from $B$B, that is $PA = 2\,PB$PA=2PB.

Show that the locus of $P$P is a circle, and find its centre and radius. (4 marks)