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If there exists exactly one point of the line `3x+4y+25=0` from which perpendicular tangesnt can be brawn to ellipse `(x^(2))/(a^(2))+y^(2)=1(agt1)`,

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Locus of point of intersecrtion fo perpendicular tangents is director circle.
If there axists exactly one such point on the line 3x+4y+25=0, then line must touch the dierector circle whose equation is `x^(2)+y^(2)=a^(2)=1`
Distance of centre of the circle from the line is 5.
So, we must have
`25=a^(2)+1`
`rArr a^(2)=24 :. a =sqrt(24)`
`:.` Electricity `sqrt(1-(1)/(24))=sqrt((23)/(24))`

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