Here,

Since the co-ordinates for the centre of the circle = (-4,3)

and it passes through the orgin i.e co ordinates are =(0,0)

NOw the distance from (-4,3) to point (0,0) will give the radius of the circle

Distance between two points (x1,y1) and (x2,y2) in the xy plane = \(\sqrt{{(x2 - x1)^2 + (y2 - y1)^2}}\)

Therefore the distance between (-4,3) and (0,0) = \(\sqrt{{(0 - (-4))^2 + (0 - 3)^2}}\)

=\(\sqrt{25}\) = 5

SInce Area of the circle = \(\pi * (radius)^2\)

= \(\pi * 5^2\)
=\(25\pi\) i. e option E

Since the circle is centered at the point (-4,3) and passes through the origin, we see that the radius is the hypotenuse of a triangle, with sides of 3 and 4, so we have a 3-4-5 right triangle with a hypotenuse of 5, which means the radius of the circle is also 5.

Thus, the area of the circle is π(5^2) = 25π,

Answer: E