Correct option (i) (b)(ii) (a)(iii)(d)
Explanation :
(i) y = mx + √(9m2 - 1) is a tangent to the hyperbola
x2/9 - y2/1 = 1
This passes through the point (3, 2). This implies
(3m - 2)2 = 9m2 - 1
⇒ -12m = -5
⇒ m = 5/12
Therefore, one tangent is 5x − 12y + 9 = 0. Also the tangent at the vertex (3, 0) passes through (3, 2). Hence, the other tangent through (3, 2) is x = 3. The chord of contact is
3x/9 - 2y/1 = 1
⇒ x - 6y = 3
Therefore, the sides of the triangle are
5x - 12y + 9 = 0
x = 3 and x - 6y = 3
Solving these equations, the vertices of the triangle are (3, 2), (3, 0) and (-5,-4/3). Hence, the area of the triangle is
(ii) The area of the triangle formed by two asymptotes and a tangent to the hyperbola
x2/a2 - y2/b2 = 1
is always constant which is equal to ab .
(iii) We know that the portion of the tangent is intercepted between the asymptote is bisected at the point of contact . In fact the asymptotes are x = ± 3y and the tangent at (3, 0) is x = 3. Hence, the tangent at (3,0) intersects the asymptotes at points (3, 1) and (3, − 1) so that the midpoint of the segment is (3, 0).