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18. Chords of...

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18. Chords of the standard hyperbola are tangents to the circle drawn on the line joining the foci as diameter. Find the locus of the point of intersection of tangents at the extremities of the chords.

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Q. Chords of the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

are tangents to the circle drawn on the line joining the foci as diameter. Find the locus of the point of intersection of the tangent at the extremities of the chords.

Q. Find the locus of the point of intersection of tangents drawn at the extremities of a normal chord of the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.

Deduce the corresponding result if the hyperbola be rectangular i.e.

x^{2} - y^{2} = a^{2} .

Q. From the points on the circle

x^{2} + y^{2} = a^{2}

, tangents are drawn to the hyperbola

x^{2} - y^{2} = a^{2}

; the locus of the middle points of the chords of contact is

Q. The locus of the point of intersection of tangents drawn at the extremities of a normal chord to the parabola

y^{2} = 4 a x

Q. Assertion :The chords of contact of the pair of tangents drawn from each point on the line

2 x + y = 4

x^{2} + y^{2} = 1

pass through the point

(1 / 2, 1 / 4)

. Reason: The locus of the point of intersection of the tangents at the extremities of any chord of a circle

C

, passing through a point P is the polar of

P

with respect to the circle

C

P

is the pole of its polar.

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