If $\alpha, \beta$ are roots of $ax^{2} + bx + c = 0$ and $ \alpha^{1} ,\beta^{1}$ are roots if $a^{1}x^{2}+b^{1}x+c^{1}=0$. Then the equation of circle having $A(\alpha ,\alpha ^{1})$ and $B(\beta ,\beta ^{1})$ as diameter is
- $\mathrm{b}\mathrm{b}^{1}(\mathrm{x}^{2}+\mathrm{y}^{2})+\mathrm{a}^{1}\mathrm{b}\mathrm{x} +\mathrm{a}\mathrm{b}^{1}\mathrm{y} +\mathrm{a}^{1}\mathrm{c}+\mathrm{a}\mathrm{c}^{1}=0$
- $\mathrm{a}\mathrm{a}^{1}(\mathrm{x}^{2}+\mathrm{y}^{2})+\mathrm{a}^{1}\mathrm{b}\mathrm{x} +\mathrm{a}\mathrm{b}^{1}\mathrm{y} +\mathrm{a}^{1}\mathrm{c}+\mathrm{a}\mathrm{c}^{1}=0$
- $\mathrm{c}\mathrm{c}^{1}(\mathrm{x}^{2}+\mathrm{y}^{2})+\mathrm{a}^{1}\mathrm{c}\mathrm{x} +\mathrm{a}\mathrm{c}^{1}\mathrm{x} +\mathrm{a}^{1}\mathrm{b}+\mathrm{a}\mathrm{b}^{1}=0$
- $\mathrm{c}\mathrm{c}^{1}(\mathrm{x}^{2}+\mathrm{y}^{2})-\mathrm{a}^{1}\mathrm{c}\mathrm{x} +\mathrm{a}\mathrm{c}^{1}\mathrm{x} -\mathrm{a}^{1}\mathrm{b}+\mathrm{a}\mathrm{b}^{1}=0$
The circle with diameter endpoints (x1, y1) and (x2, y2) is (x-x1)(x-x2) + (y-y1)(y-y2) = 0. Expanding this with roots alpha, beta and alpha', beta' leads to the equation involving coefficients a, b, c and a', b', c'.
The equation of a circle with endpoints of a diameter at A(x1, y1) and B(x2, y2) is given by the formula (x - x1)(x - x2) + (y - y1)(y - y2) = 0. Substituting the points A(alpha, alpha') and B(beta, beta') gives (x - alpha)(x - beta) + (y - alpha')(y - beta') = 0. Expanding this results in (x^2 - (alpha + beta)x + alpha*beta) + (y^2 - (alpha' + beta')x + alpha'*beta') = 0. Using Vieta's formulas for ax^2 + bx + c = 0, we know alpha + beta = -b/a and alpha*beta = c/a. Similarly for a'x^2 + b'x + c' = 0, alpha' + beta' = -b'/a' and alpha'*beta' = c'/a'. Substituting these values gives x^2 + bx/a + c/a + y^2 + b'y/a' + c'/a' = 0. Multiplying the entire equation by aa' yields aa'(x^2 + y^2) + a'bx + ab'y + a'c + ac' = 0.