Question 3.7.7: Find the equation of the central conic that passes through t...
Find the equation of the central conic that passes through the same three points P(-1, 5), Q(5, -3), and R(6, 4) of Example 6.
Substitution of the xy-coordinates of each of the three points P, Q, and R into (10) gives the linear system of three equations
Ax² + Bxy + Cy² = 1. (10)
A – 5B + 25C = 1
25A – 15B + 9C = 1
36A + 24B + 16C = 1 (11)
in the three unknowns A, B, and C. Reduction of the corresponding augmented coefficient matrix to reduced row-echelon form (Fig. 3.7.11) yields the values
A = \frac{277}{14212} , B = – \frac{172}{14212} , and C = \frac{523}{14212}.
If we substitute these coefficient values in (10) and multiply the result by the common denominator 14212, we get the desired equation
277 x² – 172 xy + 523 y² = 14212 (12)
of our central conic. The computer plot in Fig. 3.7.12 verifies that this rotated ellipse does indeed pass through all three points P, Q, and R.
