Question

Let S=\left\\{(x,y)∈\N×\N:9(x−3)^2+16(y−4)^2≤144\right\\}
and T=\left\\{(x,y)∈\R×\R:(x−7)^2+(y−4)^2≤36\right\\}.
Then n(S ⋂ T) is equal to ____ .

Answer 1

S=\left\\{(x,y)∈\N×\N:\frac{(x−3)^2}{16}+\frac{(y−4)^2}{9}≤1\right\\}
represents all the integral points inside and on the ellipse
$\frac{(x−3)^2}{16}+\frac{(y−4)^2}{9}=1,$ in first quadrant.
and T=\left\\{(x,y)∈\R×\R:(x−7)^2+(y−4)^2≤36\right\\}
represents all the points on and inside the circle
$(x−7)^2+(y−4)^2=36$

Fig.

∴(S∩T)=\left\\{(3,1)(2,2)(3,2)(4,2)(5,2)(2,3)……….(6,5)\right\\}
Total number of points = 27
So, the correct answer is 27.