Explanation
Solution
Step 1: Find Orthocentre (α,β).
Let A(1,2),B(2,3),C(3,1).
Slope of BC=3−21−3=−2. Altitude from A has slope 21. Equation: y−2=21(x−1)⟹x−2y+3=0.
Slope of AC=3−11−2=−21. Altitude from B has slope 2. Equation: y−3=2(x−2)⟹2x−y−1=0.
Solving the two altitudes: x=1,y=2. So α=1,β=2 (Note: The orthocentre is actually vertex A because the triangle is right-angled at A).
Step 2: Find the roots.
r1=α+4β=1+8=9.
r2=4α+β=4+2=6.
Step 3: Form the Quadratic Equation.
x2−(r1+r2)x+(r1⋅r2)=0.
x2−(9+6)x+(9⋅6)=0⟹x2−15x+54=0.
Correction check: Re-calculating slopes. AB slope is 1, AC slope is −0.5. It is not right-angled.
Solving x−2y+3=0 and 2x−y−1=0: x=5/3,y=7/3.
r1=5/3+28/3=11, r2=20/3+7/3=9.
x2−20x+99=0.
Correct Option: (1)
