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  • The line      intersects the ellipse <img alt="\mathrm {

The line   \mathrm { y=2 t^2}   intersects the ellipse \mathrm {\frac{x^2}{9}+\frac{y^2}{4}=1 }  in real point if
 

Option: 1

\mathrm {|t| \leq 1}


Option: 2

\mathrm {|t|<1}


Option: 3

\mathrm {|t|>1}


Option: 4

\mathrm {|t| \geq 1}


Answers (1)

best_answer

Keep \mathrm {y=2 t^2}   in    \mathrm {\frac{x^2}{9}+\frac{y^2}{4}=1, } we get
\mathrm { \frac{x^2}{9}+\frac{\left(2 t^2\right)^2}{4}=1 \Rightarrow x^2=9\left(1-t^4\right) }
If line intersects the ellipse in real points, then \mathrm { x } has real roots.


So, \mathrm {x^2 \geq 0 \Rightarrow 9\left(1-t^4\right) \geq 0 \Rightarrow\left(1-t^2\right)\left(1+t^2\right) \geq 0\\\\ }

\mathrm { \Rightarrow 1-t^2 \geq 0 \Rightarrow|t| \leq 1} . \quad\left[\because 1+\mathrm t^2>0\right]

Posted by

Devendra Khairwa

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