Question
If the point 
,lies in the region corresponding to the acute angle between the lines 2y=x and 4y=x then - .....
Hint:
When a point lies in between the lines. Then, the point satisfy that the product of values with respect to lines should be less than zero.
The correct answer is:
Given That:
If the point ,lies in the region corresponding to the acute angle between the lines 2y=x and 4y=x then:
>>> The joint equation of the two lines x=2y and x=4y becomes:
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
>>> Therefore, the range of a is (2,4).
u ≡ x - 2y = 0 and v ≡ x - 4y = 0
>>> S(x, y) ≡ x² - 6xy + 8y² = 0
>>> ( a - 2 )( a - 4 ) < 0
Related Questions to study
Consider A(0,1) and B(2,0) and P be a point on the line 4x+3y+9=0, co-ordinates of P such 
is maximum is
Hence the point is (
, 
).
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Assertion (A): The lines represented by 
and x+ y=2 do not form a triangle
Reason (R): The above three lines concur at (1,1)
Both Assertion and Reason are correct and the Reason is the correct explanation of Assertion.
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Reason (R): The above three lines concur at (1,1)
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P1,P2,P3, be the product of perpendiculars from (0,0) to 
respectively then:
P1 = 1;
P2 = 
;
P3 = 
;
>>> Therefore, we can say that P1>P2>P3.
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P1 = 1;
P2 = ;
P3 = ;
>>> Therefore, we can say that P1>P2>P3.
If θ is angle between pair of lines 
, then 
>>> 
= 2.
>>> tan
= 
>>> 
= 10.
If θ is angle between pair of lines , then
>>> = 2.
>>> tan =
>>> = 10.
If the pair of lines 
intersect on the x-axis, then 2fgh=
If the pair of lines intersect on the x-axis, then 2fgh=
If the pair of lines intersect on the x-axis, then ac=
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If the equation 
represents a pair of perpendicular lines then its point of intersection is
If the equation represents a pair of perpendicular lines then its point of intersection is
If the lines 
and 
are concurrent then λ
>>> The value of is 2.
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>>> The value of is 2.
The equation of the line concurrent with the pair of lines 
is
Hence, x=y is the the line that is concurrent with the pair of straight lines.
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Hence, x=y is the the line that is concurrent with the pair of straight lines.
If the equation 
represents a pair of straight lines then their point of intersection is
>>>The point of intersection of the pair of straight lines x2 – 5xy + 6y2 + x – 3y = 0 is (-3, -1)
If the equation represents a pair of straight lines then their point of intersection is
>>>The point of intersection of the pair of straight lines x2 – 5xy + 6y2 + x – 3y = 0 is (-3, -1)
is
is:
