If two tangents drawn from the point (al...

If two tangents drawn from the point `(alpha,beta)` to the parabola `y^2=4x` are such that the slope of one tangent is double of the other, then prove that `alpha=2/9beta^2dot`

Text Solution

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Tangent to the parabola `y^(2)=4x` having slope m is `y=mx+(1)/(m)`
It passes through `(alpha,beta)`. Therefore,
`beta=malpha+(1)/(m)`
`oralpham^(2)-betam+1=0`
According to the question, it has roots `m_(1)and2m_(1)`. Now,
`m_(1)+2m_(1)=(beta)/(alpha)andm_(1)*2m_(1)=(1)/(alpha)`
`or2((beta)/(3alpha))^(2)=(1)/(alpha)`
`oralpha=(2)/(9)beta^(2)`

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