MHB Homework: Bisectors and Intersection Points of Straight Lines

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One of the angle bisectors of the lines defined by the equation ax² + 2hxy + by² = 0 will intersect at the point of intersection of the lines represented by ax² + 2hxy + by² + 2gx + 2fy + c = 0 if the condition h(g² - f²) = fg(a - b) holds. The intersection point of the lines is identified as the center of the conic, given by specific coordinates. The angle bisector equation is hx² + (b - a)xy - hy² = 0. Substituting the intersection point into the bisector equation leads to a polynomial that simplifies to a product of factors. The condition ab - h² is always negative for conics consisting of two straight lines, confirming that the other factor must equal zero, thus proving the statement.
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Show that one of the bisectors of the angles between the pair of straight lines ax2+2hxy+by2=0 will pass through the point of intersection of the straight lines ax2+2hxy+by2+2gx+2fy+c=0 if
h(g2-f2)=fg(a-b)

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suvadip said:
Show that one of the bisectors of the angles between the pair of straight lines $ax^2+2hxy+by^2=0$ will pass through the point of intersection of the straight lines $ax^2+2hxy+by^2+2gx+2fy+c=0$ if $h(g^2-f^2)=fg(a-b).$
If the conic $ax^2+2hxy+by^2+2gx+2fy+c=0$ consists of two straight lines, then the point of intersection of those lines is the centre of the conic, which is given here to be the point $$\left(\frac{hf-bg}{ab-h^2},\frac{hg-af}{ab-h^2}\right).\qquad(1)$$ The equation of the pair of angle bisectors of the lines $ax^2+2hxy+by^2=0$ is given here to be $$hx^2 + (b-a)xy -hy^2 = 0.\qquad(2)$$ Substitute the point (1) into the equation (2), and you get $$h(hf-bg)^2 + (b-a)(hf-bg)(hg-af) -h(hg-af)^2 = 0.$$ Multiply that out, and you will find that some of the terms combine or cancel. The remaining terms can be factorised to give you $$(ab-h^2)(g^2h-f^2h + bfg - afg)=0.$$ But $ab-h^2\ne0$ because that expression is always negative for a conic consisting of two straight lines (see http://www.mathhelpboards.com/f11/pair-straight-lines-3646/). Therefore the other factor must be zero, which gives the required solution.
 
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