Text Solution
`8h^(2)-9ab``8h^(2)=9ab^(2)``8h=9ab``8h=9ab^(2)`
Answer : A
Solution : We have, `ax^(2)+2hxy+by^(2)=0` <br> Let slope of one line is m <br> `:.` Slope of another line is 2 m <br> We know that, <br> `m_(1)+m_(2)=-(2h)/(b)` <br> and `m_(1)m_(2)=a/b` <br> `:.m+2m=-(2h)/(b)` <br> `rArr 3m=(-2h)/(b)` ...(i) <br> and `m(2m)=(a)/(b)` <br> `rArr 2m^(2)=a/b` ...(ii) <br> On eliminating m , we get <br> `2((-2h)/(3b))^(2)=a/b` <br> `rArr 8h^(2)=9ab`
If the slope of one of the lines given by `ax^2 + 2hxy +by^2 = 0` is two times the other, then 8h2 = 9ab.
Explanation:
Given equation of pair of lines is
`ax^2 + 2hxy +by^2 = 0`
∴ `m_1 + m_2 = (-2h)/b` and m1m2 = `a/b`
According to the given condition,
`m_1 = 2m_2`
∴ `2m_2 + m_2 = (-2h)/b` and `2m_1m_2 = a/b`
∴ `m_2 = (-2h)/(3b)` and `m_2^2 = a/(2b)`
∴ `((-2h)/(3b))^2 = a/(2b)`
∴ `(4h^2)/(9b^2) = a/(2b)`
∴ 8h2 = 9ab
1) 4λh = ab(1 + λ)
2) λh = ab(1 + λ)2
3) 4λh2 = ab(1 + λ)2
4) None of these
Solution:
Given ax2 + 2hxy + by2 = 0 ..(i)
Let m be the slope.
Then the other slope is λm.
We know sum of slopes, m1 + m2 = -2h/b
=> m + λm = -2h/b
=> m(1 + λ) = -2h/b
=> m = -2h/b(1 + λ) …(ii)
Product of slopes, m1m2 = a/b
=> λm2 = a/b
=> m2 = a/bλ….(iii)
Squaring (ii) and equating to (iii)
4h2/b2(1 + λ)2 = a/bλ
=> 4λh2= ab(1 + λ)2
Hence option (3) is the answer.
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