First slide

Ellipse

Question

$The equation (5 x - 1)^{2} + (5 y - 2)^{2} = (λ^{2} - 2 λ + 1) (3 x + 4 y - 1)^{2} represents an ellipse if λ \in$

Moderate

A

$(0, 1)$
B

$(0, 2) - \{1\}$
C

$(1, 2)$
D

$(- 1, 0)$

Solution

The given equation is
$\begin{array}{l} {(x - \frac{1}{5})}^{2} + {(y - \frac{2}{5})}^{2} = (λ - 1)^{2} {[\frac{3 x + 4 y - 1}{\sqrt{3^{2} + 4^{2}}}]}^{2} \\ or \sqrt{{(x - \frac{1}{5})}^{2} + {(y - \frac{2}{5})}^{2}} = | λ - 1 | \frac{| 3 x + 4 y - 1 |}{\sqrt{3^{2} + 4^{2}}} \end{array}$
$\begin{array}{l} For an ellipse, S P = e P M, where 0 < e < 1 \\ \Rightarrow 0 < | λ - 1 | < 1 \\ \Rightarrow λ \in (0, 2) - {1} \end{array}$

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