0

Let $S = \left\{ {\left( {x,y} \right) \in \mathbb{N}\times \mathbb{N} :9{{\left( {x - 3} \right)}^2} + 16{{\left( {y - 4} \right)}^2} \le 144} \right\}$

$T = \left\{ {\left( {x,y} \right) \in \mathbb{R} \times \mathbb{R} :{{\left( {x - 7} \right)}^2} + {{\left( {y - 4} \right)}^2} \le 36} \right\}$

then find the value of $n\left( {S \cap T} \right)$

Set $S$ represent an ellipse $\frac{{{{\left( {x - 3} \right)}^2}}}{{16}} + \frac{{{{\left( {y - 4} \right)}^2}}}{9} \le 1$ where $(x,y)$ are Natural Numbers.

Set $T$ represent a circle ${{{\left( {x - 7} \right)}^2} + {{\left( {y - 4} \right)}^2} \le 36}$ where $(x,y)$ are Real Numbers

Not able to do this problem , how do we proceed and get the result although I have calculated 27 from the graph enter image description here

Asaf Karagila
  • 393,674
Samar Imam Zaidi
  • 8,800
  • This is doable with brute force. There are $2\cdot 3 + 1$ lattice points on the minor axis and $2\cdot 4 + 1$ on the major one (since $(x-3)^2$ should be less than $16$ and $(y-4)^2\leq 9$). In total there are $7\cdot 9=63$ possibilities which can be checked with a computer program. Most likely there will be a better solution though. – Giorgos Giapitzakis Aug 10 '22 at 13:15
  • Official answer is 27. – Samar Imam Zaidi Aug 10 '22 at 13:18
  • I didn't calculate the answer. I just said there are $63$ possibilities which you could check and find all those that work. – Giorgos Giapitzakis Aug 10 '22 at 13:20
  • To clarify, what I'm proposing is to take the $9\times 7$ rectangle that contains the ellipse and for each lattice point $(x,y)$ in that rectangle check if it fulfills your requirements ($x,y\in \mathbb{N}$, $(x,y)\in S$ and $(x,y)\in T$). A lattice point is a point with integer coordinates. – Giorgos Giapitzakis Aug 10 '22 at 13:28

1 Answers1

2

HINT.-Let the functions $f(x,y)=(x-7)^2+(y-4)^2$ and $g(x,y)=9(x-3)^2+16(y-4)^2$. We need $f(x,y)\le36$ and $g(x,y)\le144$. Extremal points are given by $f(1,4)=36,g(3,1)=g(3,7)=g(7,4)=144$ so you have

Line $y=7$: one point (by convexity of ellipse)

Line $y=6$: four points because $g(1,6)$ and $g(7,6)$ are greater than 144.

Line $y=5$: five points because $f(1,5)\gt36$ and $g(7,5)\gt144$.

Line $y=4$: seven points.

By symmetry $S\cap T$ have $$7+2(4+5+1)=27\space\text {points }$$

enter image description here

Piquito
  • 29,594