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If one of the lines of the pair a x^2+2h...

If one of the lines of the pair `a x^2+2h x y+b y^2=0` bisects the angle between the positive direction of the axes. Then find the relation for `a ,b` and `h`.

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To solve the problem, we need to find the relationship between \( a \), \( b \), and \( h \) given that one of the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \) bisects the angle between the positive direction of the axes. ### Step-by-Step Solution: 1. **Understanding the Equation of the Pair of Lines**: The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines through the origin. The lines can be expressed in the form \( y = m_1x \) and \( y = m_2x \), where \( m_1 \) and \( m_2 \) are the slopes of the lines. 2. **Identifying the Angle Bisector**: ...
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Statement 1 : If -2h=a+b , then one line of the pair of lines a x^2+2h x y+b y^2=0 bisects the angle between the coordinate axes in the positive quadrant. Statement 2 : If a x+y(2h+a)=0 is a factor of a x^2+2h x y+b y^2=0, then b+2h+a=0

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If x^2-2p x y-y^2=0 and x^2-2q x y-y^2=0 bisect angles between each other, then find the condition.

If x^2-2p x y-y^2=0 and x^2-2q x y-y^2=0 bisect angles between each other, then find the condition.

If the gradient one of the lines ax^2+2h x y+by^2=0 is twice that of the other, then h^2 =_______

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