The line \(y=3 x+4\) meets the curve \(y=2 x^{2}+8 x+1\) at two points \(A\) and \(B\).
Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(a x+b y+c=0\), where \(a, b\) and \(c\) are integers.

The line \(y=4x-3\) meets the curve \(y=3+5x-2x^2\) at the points \(A\) and \(B\).

(a) Find the coordinates of \(A\) and \(B\).

(b) The perpendicular bisector of the line \(AB\) cuts the coordinate axes at the points \(P\) and \(Q\). Given that \(O\) is the origin, find the area of triangle \(POQ\).

The straight line \(y=2 x+1\) intersects the curve \(y+x y+3 x^{2}=15\) at the points \(A\) and \(B\). The point \(C\) with coordinates \(\left(\frac{21}{10}, k\right)\) lies on the perpendicular bisector of \(A B\). (a) Find the exact value of \(k\).

(b) The point \(D\) lies on the perpendicular bisector of \(A B\) such that its perpendicular distance from \(A B\) is twice that of the point \(C\) from \(A B\). Find the possible coordinates of \(D\).

(a) The curves \(4 x^{2}-3 y^{2}+x y=24\) and \(y=\frac{2}{x}\) intersect at the points \(P\) and \(Q\). Find the coordinates of \(P\) and \(Q\). (b) Find the length of \(P Q\). Give your answer in the form \(a \sqrt{b}\), where \(a\) is rational and \(b\) is the smallest possible integer.

(a) The line \(y=3 x-2\) intersects the curve \(2 x^{2}-x y+y^{2}=2\) at the points \(A\) and \(B\). The point \(C\) with coordinates \(\left(k, \frac{7}{8}\right)\) lies on the perpendicular bisector of the line \(A B\). Find the exact value of \(k\). (b) The point \(D\) lies on the perpendicular bisector of \(A B\) such that \(D\) is a reflection of \(C\) in the line \(A B\). Find the coordinates of \(D\).

DO NOT USE A CALCULATOR IN THIS QUESTION. (a) Find the exact distance between the two points where the curve \(9(x-1)^{2}+4(y-3)^{2}=36\) cuts the \(y\)-axis. (b) Find the coordinates of the points where the curve with equation \(2 x^{2}+83 x y=x^{3} y-20 x\) intersects the curve with equation \(y=\frac{1}{x}\). Give each of your answers in the form \(a+b \sqrt{c}\), where \(a\) and \(b\) are rational and \(c\) is the smallest integer possible.

Find the value of the constant \(a\) for which the line

\(y=(2a+1)x-10\)

is a tangent to the curve

\(y=ax^2-5x+2.\)

Find the non-zero value of \(k\) for which the line \(y=-2x-6k-1\) is a tangent to the curve \(y=x(x+2k)\).

Find the exact values of \(k\) such that the straight line

\(y=1-k-x\)

is a tangent to the curve

\(y=kx^2+x+2k.\)

Do not use a calculator in this question.

Find the \(x\)-coordinates of the points where the line

\(y=3x-8\)

cuts the curve

\(y=2x^3+3x^2-26x+22.\)

Do not use a calculator in this question.

Find the \(x\)-coordinates of the points of intersection of the curves

\(y=7x^3-7x^2-17x-4\)

and

\(y=x^3-2x^2-4x-16.\)

The line \(y=kx+6\) intersects the curve \(y=x^3-4x^2+3kx+2\) at the point where \(x=2\).

(a) Find the value of \(k\).

(b) Show that the line intersects the curve at only one point.

Find the coordinates of the points of intersection of the curves

\(x^2=5y-1\)

and

\(y=x^2-2x+1.\)

Find the values of \(k\) for which the line \(y=x-3\) intersects the curve

\(y=k^2x^2+5kx+1\)

at two distinct points.

The line \(y=\frac{2}{3}x-2\) intersects the curve

\(x^2+xy=9.\)

Find the coordinates of the points of intersection.

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 \(y=3(2x+3)\).

Do not use a calculator in this question.

The line \(y=4x-6\) intersects the curve \(y=10x^3-19x^2-x\) at the points \(A\), \(B\) and \(C\). Given that \(C\) is the point \((2,2)\), find the coordinates of the midpoint of \(AB\).

Do not use a calculator in this question.

The curve \(xy=11x+5\) cuts the line \(y=x+10\) at the points \(A\) and \(B\). The midpoint of \(AB\) is the point \(C\). Show that the point \(C\) lies on the line \(x+y=11\).

Find the values of \(k\) for which the line \(y=kx+3\) does not meet the curve \(y=x^2+5x+12\).

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 line \(y=12-2x\) is a tangent to two curves. Each curve has an equation of the form

\(y=k+6+kx-x^2,\)

where \(k\) is a constant.

(i) Find the two values of \(k\).

The line \(y=12-2x\) is a tangent to one curve at the point \(A\) and the other curve at the point \(B\).

(ii) Find the coordinates of \(A\) and of \(B\).

(iii) Find the equation of the perpendicular bisector of \(AB\).

Two lines are tangents to the curve

\(y=12-4x-x^2.\)

The equation of each tangent is of the form

\(y=2k+1-kx,\)

where \(k\) is a constant.

(i) Find the two possible values of \(k\).

(ii) Find the coordinates of the point of intersection of the two tangents.

A line with equation

\(y=-5x+k+5\)

is a tangent to a curve with equation

\(y=7-kx-x^2.\)

(i) Find the two possible values of \(k\).

(ii) Find, for each of your values of \(k\), the equation of the tangent, the equation of the curve, and the coordinates of the point of contact of the tangent and the curve.

(iii) Find the distance between the two points of contact.

The curve \(3x^2+xy-y^2+4y-3=0\) and the line \(y=2(1-x)\) intersect at the points \(A\) and \(B\).

(i) Find the coordinates of \(A\) and \(B\).

(ii) Find the equation of the perpendicular bisector of the line \(AB\), giving your answer in the form \(ax+by=c\), where \(a\), \(b\) and \(c\) are integers.