If the length of the latus rectum of the ellipse $x^{2}+4 y^{2}+2 x+8 y-\lambda=0$ is 4 , and $l$ is the length of its major axis, then $\lambda+l$ is equal to ____________.
Answer (integer)
75
Solution
<p>Equation of ellipse is : ${x^2} + 4{y^2} + 2x + 8y - \lambda = 0$</p>
<p>${(x + 1)^2} + 4{(y + 1)^2} = \lambda + 5$</p>
<p>$${{{{(x + 1)}^2}} \over {\lambda + 5}} + {{{{(y + 1)}^2}} \over {\left( {{{\lambda + 5} \over 4}} \right)}} = 1$$</p>
<p>Length of latus rectum $$ = {{2\,.\,\left( {{{\lambda + 5} \over 4}} \right)} \over {\sqrt {\lambda + 5} }} = 4$$.</p>
<p>$\therefore$ $\lambda = 59$.</p>
<p>Length of major axis $= 2\,.\,\sqrt {\lambda + 5} = 16 = l$</p>
<p>$\therefore$ $\lambda + l = 75$.</p>
About this question
Subject: Mathematics · Chapter: Conic Sections · Topic: Parabola
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