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If tangents P Qa n dP R are drawn from a...

If tangents `P Qa n dP R` are drawn from a variable point `P` to thehyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1,(a > b),` so that the fourth vertex `S` of parallelogram `P Q S R` lies on the circumcircle of triangle `P Q R` , then the locus of `P` is (a) `x^2+y^2=b^2` (b) `x^2+y^2=a^2` (c) `x^2+y^2=a^2-b^2` (d) none of these

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Verified by Experts

The correct Answer is:
C
The fourth vertex of the parallelogram lies on the circumcircle.
Therefore, the parallelogram is cyclic,
i.e., the parallelogram is a rectangle,
i.e., the tangents are perpendicular.
Therefore, the locus of P is the director circle.
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CENGAGE PUBLICATION-HYPERBOLA-EXERCISES
  1. The tangent at a point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 me...

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  2. The locus of a point, from where the tangents to the rectangular hyp...

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  3. If tangents P Qa n dP R are drawn from a variable point P to thehyperb...

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  4. The number of points on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 from w...

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  5. If a ray of light incident along the line 3x+(5-4sqrt(2))y=15 gets ref...

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  6. The chord of contact of a point P w.r.t a hyperbola and its auxiliary ...

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  7. The ellipse 4x^2+9y^2=36 and the hyperbola a^2x^2-y^2=4 intersect at r...

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  8. The locus of the point which is such that the chord of contact of ta...

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  9. If x=9 is the chord of contact of the hyperbola x^2-y^2=9 then the equ...

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  10. If the tangent at point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2...

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  11. Let P(a sectheta, btantheta) and Q(asecphi , btanphi) (where theta+p...

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  12. A normal to the hyperbola (x^2)/4-(y^2)/1=1 has equal intercepts on th...

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  13. Portion of asymptote of hyperbola x^2/a^2-y^2/b^2 = 1 (between centre ...

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  14. If the angle between the asymptotes of hyperbola (x^2)/(a^2)-(y^2)/(b^...

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  15. Let any double ordinate P N P ' of the hyperbola (x^2)/(25)-(y^2)/(16)...

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  16. For hyperbola whose center is at (1, 2) and the asymptotes are paralle...

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  17. The asymptotes of the hyperbola (x^(2))/(a(1)^(2))-(y^(2))/(b(1)^(2))=...

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  18. If S=0 is the equation of the hyperbola x^2+4x y+3y^2-4x+2y+1=0 , then...

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  19. If two distinct tangents can be drawn from the point (alpha, alpha+1) ...

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  20. A hyperbola passes through (2,3) and has asymptotes 3x-4y+5=0 and 12 x...

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