First slide

Definition of a circle

Question

The equation
$λ^{2} x^{2} + (λ^{2} - 5 λ + 4) x y + (3 λ - 2) y^{2} - 8 x + 12 y - 4 = 0$
will represent a circle, if $λ =$

Moderate

A

1
B

4
C

2
D

none of these

Solution

The given equation will represent a circle, if
Coeff. of $x^{2} =$ Coeff of $y^{2}$ and Coeff. of $x y = 0 .$
$\Rightarrow λ^{2} = 3 λ - 2$ and $λ^{2} - 5 λ + 4 = 0$
$\Rightarrow λ^{2} - 3 λ + 2 = 0$ and $λ^{2} - 5 λ + 4 = 0$
$\Rightarrow (λ - 1) (λ - 2) = 0$ and $(λ - 1) (λ - 4) = 0$
$\Rightarrow λ = 1$

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