Simultaneous equations

The diagram shows part of the curve $y = 3 + \frac{4}{2x + 1}$ and the straight line $3y = 2x + 6$. Find the area of the shaded region, giving your answer in exact form.

The diagram shows the curve $y = 3x^2 - 2x + 1$ and the straight line $y = 2x + 5$ intersecting at the points $P$ and $Q$. Showing all your working, find the area of the shaded region.

Do not use a calculator in this question.

Find the coordinates of the points of intersection of the curve $y = (2x+3)^2(x-1)$ and the line $y = 3(2x+3)$.

Solve the equations

Find the values of $k$ for which the line $y = 1 - 2kx$ does not meet the curve $y = 9x^2 - (3k + 1)x + 5$.

The line $y = 2x + 5$ intersects the curve $y + xy = 5$ at the points $A$ and $B$. Find the coordinates of the point where the perpendicular bisector of the line $AB$ intersects the line $y = x$.

The curve $y = xy + x^2 - 4$ intersects the line $y = 3x - 1$ at the points $A$ and $B$. Find the equation of the perpendicular bisector of the line $AB$.

The line $2x - y + 1 = 0$ meets the curve $x^2 + 3y = 19$ at the points $A$ and $B$. The perpendicular bisector of the line $AB$ meets the $x$-axis at the point $C$. Find the area of the triangle $ABC$.

The line $4y = x + 8$ cuts the curve $xy = 4 + 2x$ at the points $A$ and $B$. Find the exact length of $AB$.

The line

cuts the curve

at the points $A$ and $B$. Find the length of the line $AB$.