Let C be a conic which is reflection of hyperbola (( x -2)2/3)-(( y -1)2/4)=-1 in the line x + y -1=0. Let locus of point of intersection of perpendicular tangents to conic C be S. From a point P (4, 2), pair of tangents P A and P B are drawn to S. Diameter of circumcircle of triangle P A B will be

Question

Let C be a conic which is reflection of hyperbola (( x -2)2/3)-(( y -1)2/4)=-1 in the line x + y -1=0. Let locus of point of intersection of perpendicular tangents to conic C be S. From a point P (4, 2), pair of tangents P A and P B are drawn to S. Diameter of circumcircle of triangle P A B will be (A) (5/2) (B) 5 (C) 10 (D) 20

Tardigrade Admin · Accepted Answer

Locus of point of intersection of perpendicular tangents will be director circle i.e.

x^{2} + (y + 1)^{2} = 4 - 3

\Rightarrow x^{2} + (y + 1)^{2} = 1

Clearly, circumcircle of

△ P A B

will be circle with diameter

PC

.

∴

Diameter

= PC = 5

Q. Let C be a conic which is reflection of hyperbola 3(x−2)2​−4(y−1)2​=−1 in the line x+y−1=0. Let locus of point of intersection of perpendicular tangents to conic C be S. From a point P(4, 2), pair of tangents PA and PB are drawn to S. Diameter of circumcircle of △PAB will be

25​

5

10

20

Locus of point of intersection of perpendicular tangents will be director circle i.e. x2+(y+1)2=4−3 ⇒x2+(y+1)2=1 Clearly, circumcircle of △PAB will be circle with diameter PC. ∴ Diameter =PC=5

$\frac{5}{2}$

Locus of point of intersection of perpendicular tangents will be director circle i.e. $x^{2} + (y + 1)^{2} = 4 - 3$
$\Rightarrow x^{2} + (y + 1)^{2} = 1$

Clearly, circumcircle of $△ P A B$ will be circle with diameter $PC$ .
$∴$ Diameter $= PC = 5$