If $$ax2+y2+2y+1=(x$$−$$2y+3)2$$ is an ellipse and a∈(b$$,∞$$), then the value of b is

$$ax2+y2+2y+1=(x$$−$$2y+3)2$$⇒$$ax2+(y+1)2=(x$$−$$2y+3)2$$⇒$$x2+(y+1)2=5a$$|x−$$2y+3$$|$$5$$ This represents ellipse if $$e=5a$$<$$1$$ or a∈(5$$,∞$$)

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$If a (x^{2} + y^{2} + 2 y + 1) = (x - 2 y + 3)^{2} is an ellipse and a \in (b, \infty), then the value of b is$

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$\begin{array}{l} a (x^{2} + y^{2} + 2 y + 1) = (x - 2 y + 3)^{2} \\ \Rightarrow a (x^{2} + (y + 1)^{2}) = (x - 2 y + 3)^{2} \\ \Rightarrow \sqrt{x^{2} + (y + 1)^{2}} = \frac{\sqrt{5}}{\sqrt{a}} \frac{| x - 2 y + 3 |}{\sqrt{5}} \\ This represents ellipse if e = \frac{\sqrt{5}}{\sqrt{a}} < 1 or a \in (5, \infty) \end{array}$

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Ellipse

Remember concepts with our Masterclasses.

If ax2+y2+2y+1=(x−2y+3)2 is an ellipse and a∈(b,∞), then the value of b is

ax2+y2+2y+1=(x−2y+3)2⇒ax2+(y+1)2=(x−2y+3)2⇒x2+(y+1)2=5a|x−2y+3|5 This represents ellipse if e=5a<1 or a∈(5,∞)

Ready to Test Your Skills?

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$If a (x^{2} + y^{2} + 2 y + 1) = (x - 2 y + 3)^{2} is an ellipse and a \in (b, \infty), then the value of b is$