6 The ellipse \(E _ { 1 }\) has equation
$$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$
\(E _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } 3
0 \end{array} \right]\) to give the ellipse \(E _ { 2 }\)
6
- Write down the equation of \(E _ { 2 }\)
6
- The ellipse \(E _ { 3 }\) has equation
$$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$
Describe the transformation that maps \(E _ { 2 }\) to \(E _ { 3 }\)
6
- Each of the lines \(L _ { A }\) and \(L _ { B }\) is a tangent to both \(E _ { 2 }\) and \(E _ { 3 }\)
\(L _ { A }\) is closer to the origin than \(L _ { B }\)
\(E _ { 2 }\) and \(E _ { 3 }\) both lie between \(L _ { A }\) and \(L _ { B }\)
Sketch and label \(E _ { 2 } , E _ { 3 } , L _ { A }\) and \(L _ { B }\) on the axes below.
You do not need to show the values of the axis intercepts for \(L _ { A }\) and \(L _ { B }\)
\includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475}
6 - Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form
$$x + y = c$$
where \(c\) is a constant.