4. For points $\mathrm { P } = ( \mathrm { x } 1 , \mathrm { y } 1 )$ and $\mathrm { Q } = ( \mathrm { x } 2 , \mathrm { y } 2 )$ of the coordinate plane, a new distance $\mathrm { d } ( \mathrm { P } , \mathrm { Q } )$ is defined by $\mathrm { d } ( \mathrm { P } , \mathrm { Q } ) = { } _ { \mid } ^ { \mid } \mathrm { x } 1 - \left. \mathrm { x } 2 \right| _ { \mid } ^ { \mid } + { } _ { \mid } \mathrm { y } 1 - \left. \mathrm { y } 2 \right| _ { \mid } ^ { \mid }$. Let $\mathrm { O } = ( 0,0 )$ and $\mathrm { A } = ( 3,2 )$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from O and A consists of the union of line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.
5.
(a) Prove that for all values of $\theta$
$$\left| \begin{array} { c c c } \sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left( \theta + \frac { 2 \pi } { 3 } \right) & \cos \left( \theta + \frac { 2 \pi } { 3 } \right) & \sin \left( 2 \theta + \frac { 4 \pi } { 3 } \right) \\ \sin \left( \theta - \frac { 2 \pi } { 3 } \right) & \cos \left( \theta - \frac { 2 \pi } { 3 } \right) & \sin \left( 2 \theta - \frac { 4 \pi } { 3 } \right) \end{array} \right| = 0 .$$
(b) Let ABC be an equilateral triangle inscribed in the circle $\mathrm { x } 2 + \underset { 2 } { 2 } = a 2$. Suppose perpendiculars from $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to the major axis of the ellipse $\mathrm { x } 2 / \mathrm { a } 2 + \mathrm { J } / \mathrm { b } 2 = 1 , ( \mathrm { a } > \mathrm { b } )$ meets the ellipse respectively at $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ so that $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ lie on the same side of the major axis as $\mathrm { A } , \mathrm { B } , \mathrm { C }$ respectively. Prove that the normals to the ellipse drawn at the points $\mathrm { P } , \mathrm { Q }$ and R are concurrent.