Question Detail
Question
Find the equation of the circle which passes through the points (4,1) and (6,5) and has its center on the line 4x+y=16.
Correct Answer
Option D is the correct answer.
Detailed Explanation
Let center be (h,k) on line 4h+k=16. Distance from center to both points equal: (h-4)²+(k-1)² = (h-6)²+(k-5)². Expand: h²-8h+16 + k²-2k+1 = h²-12h+36 + k²-10k+25. Simplify: -8h-2k+17 = -12h-10k+61 → 4h+8k = 44 → h+2k=11. With 4h+k=16, solve: from first, h=11-2k, substitute: 4(11-2k)+k=16 → 44-8k+k=16 → -7k=-28 → k=4, h=11-8=3. Center (3,4). Radius² = (3-4)²+(4-1)² = 1+9=10. Equation: (x-3)²+(y-4)²=10 → x²-6x+9 + y²-8y+16 = 10 → x²+y²-6x-8y+15=0.
Hint
Use the condition that distances from center to both points are equal.