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.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 $h(g^2-f^2)=fg(a-b).$
A pair of straight lines is a set of two lines that intersect at a point in a two-dimensional plane. They can be parallel, intersecting, or coincident.
In the coordinate plane, a pair of straight lines can be represented by the general equation Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0, where A, B, and H are coefficients and C, G, and F are constants.
The coefficient H represents the slope of the angle bisector of the pair of straight lines. It determines whether the lines are perpendicular, parallel, or intersecting at an acute or obtuse angle.
If the coefficient H is equal to 0, then the pair of straight lines are parallel. If the product of the slopes of the two lines is -1, then the pair of straight lines are perpendicular.
Yes, a pair of straight lines can also be represented by the point-slope form, slope-intercept form, or two-point form. These forms are typically used for specific scenarios or applications.