6 A rectangular garden is to have width \(x\) metres and length \(( x + 4 )\) metres.
- The perimeter of the garden needs to be greater than 30 metres. Show that \(2 x > 11\).
- The area of the garden needs to be less than 96 square metres. Show that \(x ^ { 2 } + 4 x - 96 < 0\).
- Solve the inequality \(x ^ { 2 } + 4 x - 96 < 0\).
- Hence determine the possible values of the width of the garden.
\(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0\). - Express this equation in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
- Write down:
- the coordinates of \(C\);
- the radius of the circle.
- Sketch the circle.
- A line has equation \(y = k x + 6\), where \(k\) is a constant.
- Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\).
- The equation \(\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0\) has equal roots. Show that
$$12 k ^ { 2 } - 7 k - 12 = 0$$
- Hence find the values of \(k\) for which the line is a tangent to the circle.