Answer: The correct option is (A) (2, -5).
Step-by-step explanation: We are to select the point that is on the circle described by the following equation :
![(x-2)^2+(y+3)^2=4~~~~~~~~~~~~~~~~~~~~~~~~~~(i)](https://tex.z-dn.net/?f=%28x-2%29%5E2%2B%28y%2B3%29%5E2%3D4~~~~~~~~~~~~~~~~~~~~~~~~~~%28i%29)
<u><em>Option (A) :</em></u>
(x, y) = (2, -5).
We have
![L.H.S.\\\\=(x-2)^2+(y+3)^2\\\\=(2-2)^2+(-5+3)^2\\\\=0+4\\\\=4\\\\=R.H.S.](https://tex.z-dn.net/?f=L.H.S.%5C%5C%5C%5C%3D%28x-2%29%5E2%2B%28y%2B3%29%5E2%5C%5C%5C%5C%3D%282-2%29%5E2%2B%28-5%2B3%29%5E2%5C%5C%5C%5C%3D0%2B4%5C%5C%5C%5C%3D4%5C%5C%5C%5C%3DR.H.S.)
So, the point (2, -5) lies on the circle (i). And so, option (A) is correct.
<u><em>Option (B) :</em></u>
(x, y) = (2, 0).
We have
![L.H.S.\\\\=(x-2)^2+(y+3)^2\\\\=(2-2)^2+(0+3)^2\\\\=0+9>4=R.H.S.](https://tex.z-dn.net/?f=L.H.S.%5C%5C%5C%5C%3D%28x-2%29%5E2%2B%28y%2B3%29%5E2%5C%5C%5C%5C%3D%282-2%29%5E2%2B%280%2B3%29%5E2%5C%5C%5C%5C%3D0%2B9%3E4%3DR.H.S.)
So, the point (2, 0) lies outside the circle (i). And so, option (B) is incorrect.
<u><em>Option (C) :</em></u>
(x, y) = (0, 0).
We have
![L.H.S.\\\\=(x-2)^2+(y+3)^2\\\\=(0-2)^2+(0+3)^2\\\\=4+9=13>4=R.H.S.](https://tex.z-dn.net/?f=L.H.S.%5C%5C%5C%5C%3D%28x-2%29%5E2%2B%28y%2B3%29%5E2%5C%5C%5C%5C%3D%280-2%29%5E2%2B%280%2B3%29%5E2%5C%5C%5C%5C%3D4%2B9%3D13%3E4%3DR.H.S.)
So, the point (0, 0) lies outside the circle (i). And so, option (C) is incorrect.
<u><em>Option (D) :</em></u>
(x, y) = (1, -4).
We have
![L.H.S.\\\\=(x-2)^2+(y+3)^2\\\\=(1-2)^2+(-4+3)^2\\\\=1+1](https://tex.z-dn.net/?f=L.H.S.%5C%5C%5C%5C%3D%28x-2%29%5E2%2B%28y%2B3%29%5E2%5C%5C%5C%5C%3D%281-2%29%5E2%2B%28-4%2B3%29%5E2%5C%5C%5C%5C%3D1%2B1%3C4%3DR.H.S.)
So, the point (2, -5) lies inside the circle (i). And so, option (D) is incorrect.
Thus, (A) is the correct option.