Question 1

Consider the hyperbola x 2 100 - y 2 64 = 1  with foci at S and S 1 , where S lies on the positive x -axis. Let P be a point on the hyperbola, in the first quadrant. Let &#8736; S P S 1 = &#945;

Accepted Answer

Plotting the diagram of the given value we have, We know that product of distances of any tangent from two foci = b 2 So,  &#948; &#215; &#946; sin &#945; 2 = b 2 &#8658; &#946; &#948; sin &#945; 2 9 = b 2 9 = 64 9 Now finding the greatest integer less than or equal to &#946; &#948; sin &#945; 2 9  which will be  7

Question 2

Let a and b be positive real numbers such that a &#62; 1  and  b &#60; a . Let P be a point in the first quadrant that lies on the hyperbola x 2 a 2 - y 2 b 2 = 1 . Suppose the tangent to th

Accepted Answer

Tangent at P x sec θ a - y tan θ b = 1 Passes through 1 , 0 sec θ = a Now slope of AP = 1 b sec θ a tan θ = 1 ⇒ b = tan θ Now b 2 = a 2 e 2 - 1 tan 2 θ = sec 2 θ e 2 - 1 ⇒ e 2 - 1 = sin 2 θ e 2 - 1 ∈ 0 , 1 ∵ θ ∈ 0 , π 2 e 2 ∈ 1 , 2 ⇒ 1 e 2 Now A 1 , 0 P a s e c θ , b tan θ = s e c 2 θ , tan 2 θ A P = tan 4 θ + tan 4 θ ⇒ A P = 2 tan 2 θ ∵ ∆ A P B is isosceles-right angled triangle, So, area = 1 2 A P 2 = 1 2 × 2 tan 4 θ = b 4 .

Hyperbola — Previous Year Questions & Solutions