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Find the maximum area of the ellipse (x^...

Find the maximum area of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` which touches the line `y=3x+2.`

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Area of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` is `piab`
Line y=3x+2 touches the ellipse
Here, m=3 and c=2
Using realtion, `c^(2)=a^(2)m^(2)+b^(2)` , we have
` 4=9^(2)+b^(2)`
Now, `(3a-b)^(2)ge0`
`rArr 9a^(2)+b^(2)ge6ab`
`rArr 6able4`
`rArrpi ab le (2pi)/(3)`
So, maximum area of hte ellipse is `(2pi)/(3)` sq. units.
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