- The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants.
The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
- Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation
$$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$
Hence, given that \(L\) is a tangent to \(H\),
- show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\).
The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
- Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).