PQ is the chord joining the points whose eccentric angles are φ 1 and φ 2 on the hyperbola 
= 1, If φ 1 – φ 2 = 2 α , where α is constant, prove that PQ touches the hyperbola cos 2 α 
– 
= 1.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Given hyperbola is 
= 1 ....(1)
Equation of the chord PQ to the hyperbola (1) is
cos 
– 
sin 
= cos 
⇒ 
cos α – 
sin 
= cos 
(Given φ 1 – φ 2 = 2 α )
i.e., y = 
x + 
...(2)
Comparing this line with y = mx + c
m = 
and c = 
For line y = mx + c to be tangent on 
cos 2 a – 
= 1, we have
c 2 = 
m 2 – b 2
LHS = c 2 = 
RHS = 
m 2 – b 2 = 
× 
– b 2 = 
– b 2
= 
Hence proved.
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