Homogeneous Equation

Homogeneous equation in two variable

Homogeneous equation in two variable.

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Homogeneous equation in two variable i.e $ax^2 + 2hxy +by^2 = 0$  always represent pair of line.

Since $g=f=c=0\;\Rightarrow bg^2+ch^2+af^2-2hfg-abc=0\,\forall\,x \in R$

Pair of lines represented by homogeneous equation in two variables are $L_1:y – m_1x = 0$ and $L_2 : y – m_2x = 0$, where $m_1,m_2$ are slopes of two lines that passes through origin.

Hence Equation of lines $L_1L_2=(y – m_1x)(y – m_2x) = 0$ and given that the pair of lines $L_1L_2 = ax^2+2hxy+by^2=0$.

Hence from both equations we get

$\cfrac{m_1m_2}{a}=\cfrac{- (m_1+m_2)}{2h}=\cfrac{1}{b}$

$m_1m_2=\cfrac{a}{b}$ and $m_1+m_2 = – \cfrac{2h}{b}$

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  Angle between lines represented by homogeneous equation in two variable $ax^2+2hxy+by^2=0 $ $$ $$

$m_1\,,\,m_2$ are slope of lines $L_1=0$ and $L_2=0$ respectively. Let $\theta$ be the angle between the lines then,

$tan\theta = \vert \cfrac{m_1 – m_2}{1 + m_1m_2} \vert = \vert \cfrac{2\sqrt{h^2 – ab}}{a + b} \vert\;\;since\; m_1+m_2= – \cfrac{2h}{b}\;and\;m_1m_2=\cfrac{a}{b}$

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Product of perpendiculars from a point $P(x_1 , y_1)$ on the pair of lines represented by $ax^2 + 2hxy + by^2 = 0$ $$ $$

$y = m_1x \;and \; y = m_2x$ be the two lines represented combinedly as $ax^2 + 2hxy + by^2 = 0$.

$\Rightarrow m_1 + m_2 = – \cfrac{2h}{b}\;and\; m_1m_2 = \cfrac{a}{b}$

Distance of point $P(x_1 , y_2)$ from line $y = mx$ is given by $P = \cfrac{\vert y_1 – mx_1 \vert}{\sqrt{1 + m_1^2}}$

Hence product of distance of point $P(x_1 , y_1)$ from lines is $y = m_1x\;and\;y = m_2x$ is given by

$P_1P_2 = \cfrac{\vert y_1 – m_1x_1 \vert}{\sqrt{1 + m_1^2}}\cfrac{\vert y_1 – m_2x_1 \vert}{\sqrt{1 + m_2^2}}$

$=\cfrac{\vert (y_1 – m_1x_1)(y_1 – m_2x_1) \vert}{\sqrt{1 + m_1^2}\sqrt{1 + m_2^2}}$

$= \cfrac{1}{\vert b \vert}\cfrac{\vert ax_1^2 + 2hx_1y_1 + by_1^2 \vert}{\sqrt{1 + m_1^2 + m_2^2 + m_1m_2}}$

$=\cfrac{1}{\vert b \vert}\cfrac{\vert ax_1^2 + 2hx_1y_1 + by_1^2 \vert}{\sqrt{(m_1 + m_2)^2 + (1 – m_1m_2)}}$

$=\cfrac{\vert ax_1^2 + 2hx_1y_1 + by_1^2 \vert}{\sqrt{(a – b)^2 + 4h^2}}$

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Angle bisector of lines given homogeneous equation in two variable $ax^2 + 2hxy + by^2 = 0$ $$ $$

$y = m_1x \;and \; y = m_2x$ be the two lines represented combinedly as $ax^2 + 2hxy + by^2 = 0$.

$\therefore$ Equation of bisectors are

$\cfrac{y – m_1x}{\sqrt{1 + m_1^2}} = \cfrac{y – m_2x}{\sqrt{1 + m_2^2}}\;—(1) $

$\cfrac{y – m_1x}{\sqrt{1 + m_1^2}} = – \cfrac{y – m_2x}{\sqrt{1 + m_2^2}}\;—(2) $

$\therefore$ Combined equation of bisector is

$\Bigl(\cfrac{y – m_1x}{\sqrt{1 + m_1^2}}\Bigr)^2 – \Bigl(\cfrac{y – m_2x}{\sqrt{1 + m_2^2}}\Bigr)^2 = 0$

$\therefore (1+m_2^2)(y^2+m_1^2x^2-2m_1xy) – (1+m_1^2)(y^2+m_2^2x^2-2m_2xy)=0$

$(m_1^2-m_2^2)y^2 + (m_1^2+\cancel{m_1^2m_2^2}-m_2^2-\cancel{m_1^2m_2^2})x^2-2(m_1+m_1m_2^2-m_2-m_1^2m_2)xy = 0$

$(m_2 – m_1) ((m_1+m_2)(y^2 – x^2) + 2(1 – m_1m_2)xy)=0$

Since $m_1\ne m_2\;\therefore -\cfrac{2h}{b}(y^2-x^2)-2\Bigl(\cfrac{b – a}{b}\Bigr)xy = 0$

$(x^2 – y^2)h = (a – b)xy$

$\cfrac{x^2 – y^2}{a – b} = \cfrac{xy}{h}$ $$ $$

Note :-          Let given pair of lines be $ax^2+2hxy+by^2+2gx+2fy+c=0$ intersect each other at point $A (\alpha , \beta)$ then equation angle bisector is given by

                        $\cfrac{(x – \alpha)^2 – (y – \beta)^2}{a – b} = \cfrac{(x – \alpha)(y – \beta)}{h}$

Concept of Homogenisation

Let $S(x,y)=ax^2 + 2hxy + by^2 + 2gx + 2fy + c =0$ represent one of the conic section, i.e circle, parabola, ellipse or hyperbola and the line $L=lx + my + n = 0$ intersect the curve at A and B.

Combined equation of OA and OB, where O is the origin is,

From the equation of a line $L=0$ we rewrite the equation as

$\cfrac{lx + my}{-n}=1\;——–(1)$

equation of curve $S(x,y)=0$ can be written as $ax^2 + 2hxy + by^2 + 2(gx + fy).1 + c.1^2=0\;—–(2)$

From (1) and (2) we get,

$ax^2 + 2hxy + by^2 + 2(gx + fy)\Bigl(\cfrac{lx + my}{-n}\Bigr) + c\Bigl(\cfrac{lx + my}{-n}\Bigr)^2=0$

The above equation represent equation of pair ol lines OA and OB.

Summary
Homogeneous equation in two variable as pair of lines.
Article Name
Homogeneous equation in two variable as pair of lines.
Description
A second-degree homogeneous equation in two variable always represents a pair of lines. That passes through the origin.
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