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

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Solution

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

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Ellipse

Remember concepts with our Masterclasses.

The equation (5x−1)2+(5y−2)2=λ2−2λ+1(3x+4y−1)2 represents an ellipse if λ∈

The given equation is x−152+y−252=(λ−1)23x+4y−132+422 or x−152+y−252=|λ−1||3x+4y−1|32+42 For an ellipse, SP=ePM, where 0<e<1⇒ 0<|λ−1|<1⇒ λ∈(0,2)−{1}

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