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The quadratic equation 8x² + 8xy + 2y² + 26x + 13y + 15 = 0 is the equation of two straight lines. What is the separation between these two lines?

1. 7/√5
2. 7/(2√5)
3. √7/5
4. None of these

Correct answer: 7/(2√5)

Solution

8x^2+8xy+2y^2 = 2(2x+y)^2 and 26x+13y = 13(2x+y). Let u=2x+y: 2u^2+13u+15=0 gives u=-3/2 or u=-5, i.e. parallel lines 2x+y=-3/2 and 2x+y=-5. Distance = |(-3/2)-(-5)|/sqrt(2^2+1^2) = (7/2)/sqrt5 = 7/(2sqrt5).

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