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Show that xcosalpha+asin^2alpha=p touche...

Show that `xcosalpha+asin^2alpha=p` touches the parabola `y^2=4a x` if `pcosalpha+asin^2alpha=0` and that the point of contact is `(atan^2alpha,-2atanalpha)dot`

Text Solution

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The given line is
`xcosalpha+ysinalpha=p`
`ory=-xcotalpha+p" cosec "alpha`
`:.m=-cotalphaandc=p" cosec "alpha`
Since the given line touches the parabola, we have
`c=(a)/(m)`
or cm=a
or `(pcosalpha+asin^(2)alpha=0)`
`orpcosalpha+asin^(2)alpha=0`
The point of contact is
`((a)/(cot^(2)alpha),(2a)/(cotalpha))-=(atan^(2)alpha,-2atanalpha)`
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