Simultaneous equations

A continuous curve has equation y = f(x).

The table shows corresponding values of x and y for this curve, where a and b are constants.

The trapezium rule is used, with all the y values in the table, to find an approximate area under the curve between x = 3 and x = 4

Given that this area is 17.59

show that a + 2b = 51

Given also that the sum of all the y values in the table is 97.2

find the value of a and the value of b

A circle C has equation

where k is a constant.

Find in terms of k,

the coordinates of the centre of C

the radius of C

The line with equation y = 2x – 1 intersects C at 2 distinct points.

Find the range of possible values of k.

The circle C has equation

Find

the coordinates of the centre of C,

the exact radius of C, giving your answer as a simplified surd.

The line l has equation y = 3x + k where k is a constant.

Given that l is a tangent to C,

find the possible values of k, giving your answers as simplified surds.

Solve the simultaneous equations y – 2 x – 4 = 0

A sketch of part of the curve C with equation

is shown in Figure 3. Point A lies on C and has an x coordinate equal to 2

Show that the equation of the normal to C at A is y = –2 x + 7

The normal to C at A meets C again at the point B, as shown in Figure 3.

Use algebra to find the coordinates of B.

Given the simultaneous equations 2 x + y = 1

where k is a non zero constant,

show that

Given that x² + 8kx + k = 0 has equal roots,

find the value of k.

For this value of k, find the solution of the simultaneous equations.

Figure 2.

Find the coordinates ofA and the coordinates of B. (6)

Find the distance AB in the form r \2 wherer is a rational number. (3)

Figure 1 shows a sketch of the curve with equation y = , x 0 The curve C has equation y = − 5, x 0, and the line l has equation y = 4x + 2

Sketch and clearly label the graphs of C and l on a single diagram. On your diagram, show clearly the coordinates of the points where C and l cross the coordinate axes.

Write down the equations of the asymptotes of the curve C.

Find the coordinates of the points of intersection of y = − 5 and y = 4x + 2

On the axes below sketch the graphs of

y = x (4 – x)

y = x² (7 – x) showing clearly the coordinates of the points where the curves cross the coordinate axes.

Show that the x-coordinates of the points of intersection of

are given by the solutions to the equation x(x² – 8x + 4) = 0

The point A lies on both of the curves and the x and y coordinates of A are both positive.

Find the exact coordinates of A, leaving your answer in the form (p q√3, r + s√3), where p, q, r and s are integers.

Solve the simultaneous equations

The curve C has equation y=y = and the line l has equation y = 2x + 5.

On the axes below, sketch the graphs of C and l, indicating clearly the coordinates of any intersections with the axes.

Find the coordinates of the points of intersection of C and l.