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If the tangents to the parabola y^2=4a x...

If the tangents to the parabola `y^2=4a x` intersect the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` at `Aa n dB` , then find the locus of the point of intersection of the tangents at `Aa n dBdot`

Text Solution

Verified by Experts

The correct Answer is:
`y^(2)=-(b^(4))/(a^(3))x`
Tangents to parabola intersect the hyperbola at A and B.
Let the point of intersection of tangents at A and B be P(h, k).
So, AB will be chord of contact of hyperbola w.r.t. point P.
Thus, equation of AB is
`(hx)/(a^(2))-(ky)/(b^(2))=1`
`"or "(ky)/(b^(2))=(hx)/(a^(2))-1`
`"or "y=((b^(2)h)/(ka^(2)))x-(b^(2))/(k)`
This line touches the parabola.
`"So, "-(b^(2))/(k)=(a)/((b^(2)h)/(ka^(2)))" (as y = mx + c touches the parabola "y^(2)="if c=a/m)" `
`rArr" "-(b^(2))/(k)=(ka^(3))/(b^(2)h)`
Hence, required locus is `y^(2)=-(b^(4))/(a^(3))x.`
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