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Q.)A line parallel to the straight line 2x - y = 0 is tangent to hyperbola P at point (x1 , y1). Then x1^2 + 5y1^2 is equal to :

a) 6

b) 8

c) 10

d) 5

where P : 

Solution: Let's draw a sketch for hyperbola P and line:

Tangent to point (x1, y1) to P is : 

x.x1/4 - y.y1/2 = 1

As this is parallel to given line both have same slope. So,

x1/2y1 = 2  --->   x1 = 4y1                           --------- (1)

and also point (x1 , y1) also satisfy P.  So,

x1^2 / 4 - y1^2 / 2 = 1                                --------- (2)

From above (1) and (2) :

y1^2  =  2/7

As, we have to find : x1^2 + 5y1^2 

which now becomes 21y1^2  using (1).

So, answer is 21*2/7 = 6.

Option (A) is correct.

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