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Find the range of (alpha,2+alpha) and ((...

Find the range of `(alpha,2+alpha)` and `((3alpha)/2,a^2)` lie on the opposite sides of the line `2x+3y=6.`

Text Solution

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The correct Answer is:
`alpha in (-oo, -2) uu (0,1)`
We have L(x,y) = 2x+3y-6
` therefore L(alpha, 2+alpha) = 5alpha`
`" and " L((3)/(2) alpha, alpha^(2)) = 3alpha + 3alpha^(2)-6`
Given points lie on the opposite sides of the line if
`5alpha (3alpha + 3alpha^(2)-6) lt 0`
`rArr alpha (alpha + 2)(alpha-1) lt 0`
`rArr alpha in (-oo, -2) uu (0,1)`
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