Explanation
To find the diameter of a circle passing through points A(0,2), B(2,3), and D(0,17), we first find the equation of the circle.
Step 1: General equation of the circle
Let the equation be x2+y2+2gx+2fy+c=0.
Since A(0,2) lies on it: 02+22+2g(0)+2f(2)+c=0⟹4+4f+c=0⟹c=−4−4f.
Since D(0,17) lies on it: 02+172+2g(0)+2f(17)+c=0⟹289+34f+c=0.
Substitute c: 289+34f−4−4f=0⟹285+30f=0⟹f=−9.5.
Then c=−4−4(−9.5)=−4+38=34.
Step 2: Find g using point B(2,3)
22+32+2g(2)+2f(3)+c=0
4+9+4g+6(−9.5)+34=0
13+4g−57+34=0
4g−10=0⟹g=2.5.
Step 3: Calculate the radius and diameter
The center of the circle is (−g,−f)=(−2.5,9.5) and the radius R is:
R=g2+f2−c=(−2.5)2+(9.5)2−34
R=6.25+90.25−34=62.5
R=2125=255=2510
The diameter is 2R:
Diameter=2×2510=510
Correct Option: (b) 510
