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Show that xcosalpha+ysinalpha=p touches ...

Show that `xcosalpha+ysinalpha=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

Verified by Experts

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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Knowledge Check

  • If the line x+y =1 is a tangent to the parabola y^(2)-y +x=0 , then the point of contact is-

    A
    `(0, 1)`
    B
    `(a, 0)`
    C
    `(1, 1)`
    D
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