If one of varying central conic (hyperbola) is fixed in magnitude and position, prove that the locus of the point of contact of a tangent drawn to it

Question

If one of varying central conic (hyperbola) is fixed in magnitude and position, prove that the locus of the point of contact of a tangent drawn to it

If one of varying central conic (hyperbola) is fixed in magnitude and position, prove that the locus of the point of contact of a tangent drawn to it from a fixed point on the other axis is a parabole.

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

Consider hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1,` where a is fixed.
Let the coordinate of point P on the hyperbola be `(a sec theta, b tan theta)-=(h,k)`.
Equation of tangent to hyperbola at point P is
`(x sec theta)/(2)-(y tan theta)/(b)=1" (1)"`
Let this tangent pass through fixed point (0, c) on conjugate axis.
`therefore" "-c tan theta=b" (2)"`
From a sec `theta=h and b tan theta=k` and using (2), we get
`(k)/(c)=-tan^(2)theta and (h^(2))/(a^(2))=sec^(2)theta`
Adding, we get
`(k)/(c)+(h^(2))/(a^(2))=1`
`rArr" "h^(2)=(a^(2))/(c)(c-k)`
Hence, required locus is `x^(2)=-(a^(2))/(c)(y-c)`, which is a parabola.