7 The equation of a circle is \(( x - 6 ) ^ { 2 } + ( y + a ) ^ { 2 } = 18\). The line with equation \(y = 2 a - x\) is a tangent to the circle.
- Find the two possible values of the constant \(a\).
- For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent.
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The diagram shows a symmetrical plate \(A B C D E F\). The line \(A B C D\) is straight and the length of \(B C\) is 2 cm . Each of the two sectors \(A B F\) and \(D C E\) is of radius \(r \mathrm {~cm}\) and each of the angles \(A B F\) and \(D C E\) is equal to \(\frac { 1 } { 3 } \pi\) radians. - It is given that \(r = 0.4 \mathrm {~cm}\).
- Show that the length \(\mathrm { EF } = 2.4 \mathrm {~cm}\).
- Find the area of the plate. Give your answer correct to 3 significant figures.
- It is given instead that the perimeter of the plate is 6 cm .
Find the value of \(r\). Give your answer correct to 3 significant figures.