The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, x2 sec 2 α- y2 operatornamecosec2 α=1, α ∈(0, π / 4), is

Question

The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, x2 sec 2 α- y2 operatornamecosec2 α=1, α ∈(0, π / 4), is (A) 0 (B) 1 (C) 2 (D) infinite

Tardigrade Admin · Accepted Answer

(x2/ cos 2 α)-(y2/ sin 2 α)=1 Locus of perpendicular tangents is director circle. x2+y2 =a2-b2 or x2+y2 = cos 2 α- sin 2 α= cos 2 α But 0 < α < (π/4) 0 < 2 α < (π/2) So there are infinite points.

Q. The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, $x^{2} sec^{2} α -$ $y^{2} cosec^{2} α = 1, α \in (0, π /4)$ , is

0

1

2

infinite

$\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$
Locus of perpendicular tangents is director circle.
$x^{2} + y^{2} = a^{2} - b^{2}$
or $x^{2} + y^{2} = cos^{2} α - sin^{2} α = cos 2 α$
But $0 < α < \frac{π}{4}$
$0 < 2 α < \frac{π}{2}$
So there are infinite points.

Q. The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, x2sec2α− y2cosec2α=1,α∈(0,π/4), is

0

1

2

infinite

cos2αx2​−sin2αy2​=1 Locus of perpendicular tangents is director circle. x2+y2=a2−b2 or x2+y2=cos2α−sin2α=cos2α But 0<α<4π​ 0<2α<2π​ So there are infinite points.

Q. The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, $x^{2} sec^{2} α -$ $y^{2} cosec^{2} α = 1, α \in (0, π /4)$ , is

$\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$
Locus of perpendicular tangents is director circle.
$x^{2} + y^{2} = a^{2} - b^{2}$
or $x^{2} + y^{2} = cos^{2} α - sin^{2} α = cos 2 α$
But $0 < α < \frac{π}{4}$
$0 < 2 α < \frac{π}{2}$
So there are infinite points.