If we rotate axes through an angle 45º in the clockwise direction, the equation of the rectangular hyperbola x2 – y2 = a2 reduces to xy = 
to xy = c2 (writing c2 for 
). This hyperbola is easier to handle. Any point on this hyperbola may be taken as 
.
On the basis of above passage, answer the following questions.
(i)The asymptotes to the hyperbola xy = c2 must be
Text Solution
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Ans.
(i)
Sol. The line y = x and c y = –x meets the curve xy = c2 at a finite distance from origin.
⇒ they cannot be asymptotes.
⇒ is false.
Similarly is ruled out.
y = 
, x = 
as x → ∞ , y → 0.
Also as y → ∞ , x → 0
⇒ The lines y = 0 and x = 0 are asymptotes to the curve.
xy = c2
⇒ is the correct answer.
(ii)
Sol. xy = c2 ⇒ y = 
, 
= 
⇒ Slope of normal at 
= 
= –t2
⇒ Equation of normal at 
must be y – 
= –t2
(x – ct) etc.
(iii)
Sol. From Q.2 the equation of normal at 
is
t3x – ty – ct4 + c = 0
Put x = ct1, y = 
, we get t3.ct1 – t. 
–c. t4 + c = 0
⇒ (t1 – t) (t3t1 – 1) = 0 ⇒ t1 = 
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