Consider the sets A=\ (x, y) ∈ ℝ × ℝ: x^2+y^2=25\ , B=\ (x, y) ∈ ℝ...

\mathrm{A}: \mathrm{x}^2+\mathrm{y}^2=25...(i)$<br/>B :$\frac{x^2}{144}+\frac{y^2}{16}=1...(ii)$<br/>C:$x^2+y^2 \leq 4...(iii)$<br/>Solve (1) \& (2)<br/>$$\begin{aligned}<br/>& x^2+9\left(25-x^2\right)=144 \\ & -8 x^2=144-225=-81 \\ & x= \pm \frac{9}{2 \sqrt{2}}<br/>\end{aligned}$$<br/>$\operatorname{By}(1) \Rightarrow y= \pm \sqrt{25-x^2}$<br/>$= \pm \sqrt{25-\frac{81}{8}}= \pm \frac{\sqrt{119}}{2 \sqrt{2}}$<br/>$\therefore $\mathrm{D}=\mathrm{A}$ \cap \mathrm{~B}=<br/>\left\{\left(\frac{9}{2 \sqrt{2}}, \frac{\sqrt{119}}{2 \sqrt{2}}\right),\left(\frac{9}{2 \sqrt{2}},-\frac{\sqrt{119}}{2 \sqrt{2}}\right),\left(\frac{-9}{2 \sqrt{2}}, \frac{\sqrt{119}}{2 \sqrt{2}}\right),\left(\frac{-9}{2 \sqrt{2}}, \frac{-\sqrt{119}}{2 \sqrt{2}}\right)<br/>No. of elements in setD=4$<br/><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/hiM-DfQup4kp5DeFXCKLYfKiRDKdXweZmUqd6gSXHx8.original.fullsize.png"/><br/>$$\begin{aligned}<br/>& \because C=\left\{(x, y) \in \mathrm{Z} \times \mathrm{Z}:\mathrm{x}^2+\mathrm{y}^2\leq 4\right\}\\ & =\{(0,2),(2,0),(0,-2),(-2,0),(1,1),(-1,-1), \\ & (1,-1),(-1,1),(1,0),(0,1),(-1,0),(0,-1), \\ & (0,0)\}<br/>\end{aligned}$$<br/>No.$ of elements in set \mathrm{C}=1$3$
Total no. of one-one function from
Set $D to \sec C \Rightarrow 13 \times 12 \times 11 \times 10=17160