Question

Please help! A-level maths.

Answer 1

`x+y=k\implies y=k-x`

Substitute this into the parabolic equation,

`k-x=6-(x-3)^2\implies x^2-7x+(k+3)=0`

We're told the line
`x+y=k` intersects
`C` twice, which means the quadratic above has two distinct real solutions. Its discriminant must then be positive, so we know

`(-7)^2-4(k+3)=49-4(k+3)>0\implies k<\frac{37}4=9.25`

We can tell from the quadratic equation that
`C` has its vertex at the point (3, 6). Also, note that

`-1\le\sin t\le1\implies3\le3+2\sin t\le5\implies3\le x\le5`

and

`-1\le\cos2t\le1\implies2\le4+2\cos2t\le6\implies2\le y\le6`

so the furthest to the right that
`C` extends is the point (5, 2). The line
`x+y=k` passes through this point for
`2+5=k\implies k=7`. For any value of
`k<7`, the line
`x+y=k` passes through
`C` either only once, or not at all.

So
`7\le k<9.25`; in set notation,

`\{k\mid 7\le k<9.25\}`