Question

Let \(Q\) denote the quarter-disc of points \((x,y)\) such that \(x\geq 0\), \(y\geq 0\) and \(x^2+y^2\leq1\) as drawn in Figures A and B below.

Set of axes with x and y both marked from 0 to 2, and a quarter-disc of radius 1 centre the origin shaded.
Figure A
The same as figure A.
Figure B
  1. On the axes in Figure A, sketch the graphs of \[x+y=\frac{1}{2}, \qquad x+y=1, \qquad x+y=\frac{3}{2}.\]

    What is the largest value of \(x+y\) achieved at points \((x,y)\) in \(Q\)? Justify your answer.

  2. On the axes in Figure B, sketch the graphs of \[xy=\frac{1}{4}, \qquad xy=1, \qquad xy=2.\]

    What is the largest value of \(x^2+y^2+4xy\) achieved at points \((x,y)\) in \(Q\)? What is the largest value of \(x^2+y^2-6xy\) achieved at points \((x,y)\) in \(Q\)?

  3. Describe the curve \[x^2+y^2-4x-2y=k\] where \(k>-5\).

    What is the smallest value of \(x^2+y^2-4x-2y\) achieved at points \((x,y)\) in \(Q\)?