1 Equation 2 variables
The number of real values of (a,b) which satisfy the equation
a − b = a × b is?
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4 solutions
Can you check your solution?
I do not get tan 4 θ − sin 4 θ = tan 2 θ sin 2 θ as an identity.
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What is the problem?
He means tan 2 θ − sin 2 θ = tan 2 θ sin 2 θ . @Krishna Sharma can you explain why it satisfies all values of θ ? Very interesting question :)
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Simplifing
c o s 2 x s i n 2 x − s i n 2 x
Taking LCM
c o s 2 x s i n 2 x ( 1 − c o s 2 x )
1 − c o s 2 x = s i n 2 x
c o s 2 x s i n 4 x
Which equals
t a n 2 x s i n 2 x
the no of real roots is 2 not infinite
a-b=a*b
⟹
1/b- 1/a =1
⟹
1/b=1+1/a.
For any value of `
b
=
0
, we can always find an a.
Why is that? Can you explain?
First divide ab on both sides, we get 1/b-1/a = 1.
Trial and error with possible pairs can eliminate the wrong options. The following pairs of (a,b) satisfies the above equation: (0,0), (1,1/2), (1/3,1/2). So the answer cannot be 0 or 1 or 2. It is more than 2.
a-b=ab》》a=ab+b》》a=b (a+1)》》a/(a+1)=b So as long as b = a/a+1 the equation is true, obviously the answer is infinity
Put a = tan 2 θ
b = sin 2 θ
Simplifying
c o s 2 x s i n 2 x − s i n 2 x
Taking LCM
c o s 2 x s i n 2 x ( 1 − c o s 2 x )
1 − c o s 2 x = s i n 2 x
c o s 2 x s i n 4 x
Which equals
t a n 2 x s i n 2 x
which satifies the equation for all values of θ hence the answer is infinite