If a tangent to the parabola `y^2 = 4ax` intersects the `x^2/a^2+y^2/b^2= 1` at `A `and `B`, then the locus of the point of intersection of tangents a

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If a tangent to the parabola `y^2 = 4ax` intersects the `x^2/a^2+y^2/b^2= 1` at `A `and `B`, then the locus of the point of intersection of tangents a

If a tangent to the parabola `y^2 = 4ax` intersects the `x^2/a^2+y^2/b^2= 1` at `A `and `B`, then the locus of the point of intersection of tangents at `A` and `B` to the ellipse is

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - `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.`