Tangency condition for line and curve

5 The equation of a curve is \(y = 4 x ^ { 2 } - k x + \frac { 1 } { 2 } k ^ { 2 }\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

1. Given that the curve and the line intersect at the points with \(x\)-coordinates 0 and \(\frac { 3 } { 4 }\), find the values of \(k\) and \(a\).
2. Given instead that \(a = - \frac { 7 } { 2 }\), find the values of \(k\) for which the line is a tangent to the curve. [5]

7 The straight line \(\mathrm { y } = \mathrm { x } + 5\) meets the curve \(2 \mathrm { x } ^ { 2 } + 3 \mathrm { y } ^ { 2 } = \mathrm { k }\) at a single point \(P\).

1. Find the value of the constant \(k\).
2. Find the coordinates of \(P\).

1. The curve \(C\) has equation

$$y = \frac { k ^ { 2 } } { x } + 1 \quad x \in \mathbb { R } , x \neq 0$$ where \(k\) is a constant.

1. Sketch \(C\) stating the equation of the horizontal asymptote. The line \(l\) has equation \(y = - 2 x + 5\)
2. Show that the \(x\) coordinate of any point of intersection of \(l\) with \(C\) is given by a solution of the equation $$2 x ^ { 2 } - 4 x + k ^ { 2 } = 0$$
3. Hence find the exact values of \(k\) for which \(l\) is a tangent to \(C\).

1. A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 14 y = 40\).

The line \(l\) has equation \(y = x + k\), where \(k\) is a constant.
a. Show that the \(x\)-coordinate of the points where \(C\) and \(l\) intersect are given by the solutions to the equation $$2 x ^ { 2 } + ( 2 k - 20 ) x + k ^ { 2 } - 14 k - 40 = 0$$ b. Hence find the two values of \(k\) for which \(l\) is a tangent to \(C\).

1. A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2 x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
2. find the two possible values of \(r\), giving your answers as fully simplified surds.

5 In this question you must show detailed reasoning.
The line \(x + 5 y = k\) is a tangent to the curve \(x ^ { 2 } - 4 y = 10\). Find the value of the constant \(k\).

8 A line has equation \(y = m x - 1\), where \(m\) is a constant.
A curve has equation \(y = x ^ { 2 } - 5 x + 3\).

1. Show that the \(x\)-coordinate of any point of intersection of the line and the curve satisfies the equation $$x ^ { 2 } - ( 5 + m ) x + 4 = 0$$
2. Find the values of \(m\) for which the equation \(x ^ { 2 } - ( 5 + m ) x + 4 = 0\) has equal roots.
   (4 marks)
3. Describe geometrically the situation when \(m\) takes either of the values found in part (b).
   (1 mark)