Point \(A\) has coordinates \((3,-1)\).
A circle has equation \((x-4)^2+(y+3)^2=5\).
(a) Show that \(A\) lies on the circumference of the circle.
(b) Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\).
(c) Find the equation of the tangent to the circle at \(A\).
A circle with centre \(C\) has the equation \(x^2+y^2-10x-4y+24=0\).
(a) Show that the line \(y=2x-3\) is a tangent to this circle.
(b) Given that this tangent touches the circle at the point \(P\), find the coordinates of \(P\).
(c) Find the equation of the circle which has its centre at \(P\) and passes through the origin.
Solutions by accurate drawing will not be accepted. A circle, \(C\), has equation \((x-5)^{2}+(y-12)^{2}=100\). (a) Find the equation of the tangent to \(C\) at the point \((11,4)\).
Give your answer in the form \(a x+b y=c\), where \(a, b\) and \(c\) are integers. (b) Show that \(C\) and the circle with equation \(x^{2}+y^{2}=4\) do not intersect.
The point \(A\) has coordinates \((-3,6)\).
The point \(B\) has coordinates \((7,-8)\).
Given that the line \(AB\) is the diameter of a circle, find the equation of the circle.
A circle has equation \(x^{2}+y^{2}-25=0\).
A second circle has the same radius as the first circle, and the coordinates of its centre are both positive.
The two circles intersect at the points \(A\) and \(B\).
The line \(A B\) has length 6 and is parallel to the line \(y=-x\).
Find the equation of the second circle in the form \(x^{2}+y^{2}+a x+b y+c=0\), where \(a, b\) and \(c\) are constants.
The lines \(x=0\), \(x=4\), \(y=3\) and \(y=-1\) are tangents to a circle.
(a) Find the equation of the circle.
The line \(y=2x+a\), where \(a\) is a constant, is also a tangent to the circle.
(b) Show that \(5x^2+4(a-2)x+(a-1)^2=0\), and hence find the possible values of \(a\). Give your answers in exact form.
The line with equation
\(x+3y=k,\)
where \(k\) is a positive constant, is a tangent to the curve with equation
\(x^2+y^2+2y-9=0.\)
Find the value of \(k\) and hence find the coordinates of the point where the line touches the curve.
The curves
\(y=x^2 \quad\text{and}\quad y^2=27x\)
intersect at \(O(0,0)\) and at the point \(A\). Find the equation of the perpendicular bisector of the line \(OA\).
Find the coordinates of the points where the line \(2y-3x=6\) intersects the curve
\(\frac{x^2}{4}+\frac{y^2}{9}=5.\)
The line \(y=kx+3\), where \(k\) is a positive constant, is a tangent to the curve \(x^2-2x+y^2=8\) at the point \(P\).
(i) Find the value of \(k\).
(ii) Find the coordinates of \(P\).
(iii) Find the equation of the normal to the curve at \(P\).
Differentiate \(\frac{2x^3 + 5}{x}\) with respect to \(x\).
Differentiate \(4x + \frac{6}{x^2}\) with respect to \(x\).
The function f is defined by \(f(x) = 2 - \frac{3}{4x-p}\) for \(x > \frac{p}{4}\), where \(p\) is a constant.
Find \(f'(x)\) and hence determine whether \(f\) is an increasing function, a decreasing function or neither.
A function \(f\) is defined by \(f : x \mapsto x^3 - x^2 - 8x + 5\) for \(x < a\). It is given that \(f\) is an increasing function. Find the largest possible value of the constant \(a\).
The function \(f\) is such that \(f(x) = x^3 - 3x^2 - 9x + 2\) for \(x > n\), where \(n\) is an integer. It is given that \(f\) is an increasing function. Find the least possible value of \(n\).
(i) Express \(3x^2 - 6x + 2\) in the form \(a(x+b)^2 + c\), where \(a, b\) and \(c\) are constants.
(ii) The function \(f\), where \(f(x) = x^3 - 3x^2 + 7x - 8\), is defined for \(x \in \mathbb{R}\). Find \(f'(x)\) and state, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.
The function f is defined by \(f(x) = \frac{1}{x+1} + \frac{1}{(x+1)^2}\) for \(x > -1\).
The function g is defined by \(g(x) = \frac{1}{x+1} + \frac{1}{(x+1)^2}\) for \(x < -1\).
(i) Express \(9x^2 - 12x + 5\) in the form \((ax + b)^2 + c\).
(ii) Determine whether \(3x^3 - 6x^2 + 5x - 12\) is an increasing function, a decreasing function or neither.
The equation of a curve is \(y = x^3 + ax^2 + bx\), where \(a\) and \(b\) are constants.
(i) In the case where the curve has no stationary point, show that \(a^2 < 3b\).
(ii) In the case where \(a = -6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\).
A function \(f\) is defined by \(f(x) = \frac{5}{1 - 3x}\), for \(x \geq 1\).
(i) Find an expression for \(f'(x)\).
(ii) Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.