Circle sector area and angle

7
\includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813} The diagram shows the circular cross-section of a uniform cylindrical log with centre \(O\) and radius 20 cm . The points \(A , X\) and \(B\) lie on the circumference of the cross-section and \(A B = 32 \mathrm {~cm}\).

1. Show that angle \(A O B = 1.855\) radians, correct to 3 decimal places.
2. Find the area of the sector \(A X B O\). The section \(A X B C D\), where \(A B C D\) is a rectangle with \(A D = 18 \mathrm {~cm}\), is removed.
3. Find the area of the new cross-section (shown shaded in the diagram).

9 The diagram shows a parabola \(P\) which has equation \(y = \frac { 1 } { 8 } x ^ { 2 }\), and another parabola \(Q\) which is the image of \(P\) under a reflection in the line \(y = x\). The parabolas \(P\) and \(Q\) intersect at the origin and again at a point \(A\).
The line \(L\) is a tangent to both \(P\) and \(Q\).
\includegraphics[max width=\textwidth, alt={}, center]{7441c4e6-5448-483b-b100-f8076e7e6cd8-5_1015_1089_623_479}

1. 1. Find the coordinates of the point \(A\).
   2. Write down an equation for \(Q\).
   3. Give a reason why the gradient of \(L\) must be - 1 .
2. 1. Given that the line \(y = - x + c\) intersects the parabola \(P\) at two distinct points, show that $$c > - 2$$
   2. Find the coordinates of the points at which the line \(L\) touches the parabolas \(P\) and \(Q\).
      (No credit will be given for solutions based on differentiation.)