First slide

Definition of a circle

Question

The two conics $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} = c_{1}$ and
$a_{2} x^{2} + 2 h_{2} x y + b_{2} y^{2} = c_{2}$ intersect in 4 concyclic points. Then

Moderate

A

$(a_{1} - b_{1}) h_{2} = (a_{2} - b_{2}) h_{1}$
B

$(a_{1} - b_{1}) h_{1} = (a_{2} - b_{2}) h_{2}$
C

$(a_{1} + b_{1}) h_{2} = (a_{2} + b_{2}) h_{1}$
D

$(a_{1} + b_{1}) h_{1} = (a_{2} + b_{2}) h_{2}$

Solution

The equation of a conic passing through the intersection of the given conics is
$(a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} - c_{1}) + λ (a_{2} x^{2} + 2 h_{2} x y + b_{2} y^{2} - c_{2}) = 0$
This equation will represent a circle, if
Coeff. of $x^{2} =$ Coeff. of $y^{2}$ and Coeff. of $x y = 0$
$\Rightarrow (a_{1} + λ u_{2}) = (b_{1} + λ b_{2}) and 2 h_{1} + 2 λ h_{2} = 0$
Eliminating $λ$ , from these two equations, we get
$(a_{1} - b_{1}) = \frac{h_{1}}{h_{2}} (a_{2} - b_{2}) \Rightarrow (a_{1} - b_{1}) h_{2} = (a_{2} - b_{2}) l_{1}$

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