Question

If the point P(a²,a) lies in the region corresponding to the acute angle between the lines 2y = x and 4y=x, then

(a)a = (2,4) (b)a = (2,6) (c)a = (4,6) (d)a = (4,8)

Answer 1

The joint equation of the two lines

x = 2y and x = 4y, i.e.,

u ≡ x - 2y = 0 ... and ... v ≡ x - 4y = 0 is

u·v = 0, i.e.,

( x - 2y )·( x - 4y ) = 0. i.e.,

x² - 4xy - 2xy + 8y² = 0, i.e.,

S(x,y) ≡ x² - 6xy + 8y² = 0....... (1)

If the point P(a², a) lies in the interior of the

acute angle formed by these lines, then the

value of S(x,y) at P( x=a², y=a ) is negative, i.e.,

S(a², a) ≡ (a²)² - 6(a²)(a) + 8a² < 0

∴ a² ( a² - 6a + 8 ) < 0

∴ a² - 6a + 8 < 0

∴ ( a - 2 )( a - 4 ) < 0

∴ 2 < a < 4 , i.e.,
a ∈ ( 2, 4 )