The equation 8 x^2-24 x y+18 y^2-6 x+9 y-5=0 | vedclass

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(B) The given equation is $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$.We can rewrite the quadratic part as $(2 \sqrt{2} x - 3 \sqrt{2} y)^2 = 2(4 x^2 - 12 x y +

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pair of parallel lines

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(B) The given equation is $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$.We can rewrite the quadratic part as $(2 \sqrt{2} x - 3 \sqrt{2} y)^2 = 2(4 x^2 - 12 x y + 9 y^2) = 2(2 x - 3 y)^2$.Let $2 x - 3 y = t$. Then the equation becomes $2 t^2 - 3 t - 5 = 0$.Factoring the quadratic: $2 t^2 - 5 t + 2 t - 5 = 0 \implies t(2 t - 5) + 1(2 t - 5) = 0 \implies (t + 1)(2 t - 5) = 0$.Substituting $t = 2 x - 3 y$ back,we get $(2 x - 3 y + 1)(2(2 x - 3 y) - 5) = 0$.This simplifies to $(2 x - 3 y + 1)(4 x - 6 y - 5) = 0$.Since the equation represents two linear factors of the form $(a_1 x + b_1 y + c_1)(a_2 x + b_2 y + c_2) = 0$,it represents a pair of lines.Comparing the slopes,$m_1 = \frac{2}{3}$ and $m_2 = \frac{4}{6} = \frac{2}{3}$.Since the slopes are equal,the lines are parallel.

The equation $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ represents a

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