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Construction of Incircle

3 x 2 -8xy-3 ...

Question

$3 x^{2} - 8 x y - 3 y^{2} + 10 x + 20 y - 25 = 0$ are the bisectors of angle between two lines $l_{1}$ and $l_{2}$ one of which passes through origin. Determine the equation of the other line.

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Solution

On factorising the equations of bisectors are $x - 3 y + 5 = 0$ and $3 x + y - 5 = 0$ which intersect at $(1, 2)$ . Both the lines will pass through $(1, 2)$ and one line passes through origin and hence its equation is $y = 2 x$ or $2 x - y = 0$ . The other line through $(1, 2)$ may be taken as
$y - 2 = m (x - 1)$
or $m x - y + (2 - m) = 0.... (1)$
In order to find $m$ ,we choose a point on any bisector say $3 x + y + 5 = 0$ say $(0, 5)$ . Its distance from both the lines is same
$\frac{0 - 5}{\sqrt{5}} = \pm \frac{+ 5 + 2 - m}{\sqrt{1 + m^{2}}}$
$∴ 5 (1 + m^{2}) = {(m + 3)}^{2}$
or $2 m^{2} - 3 m - 2 = 0$
or $(m - 2) (2 m + 1) = 0$ $m = 2, - 1 / 2)$
But $m = 2$ corresponds to the line $2 x - y = 0$ . hence $m = - 1 / 2$ is the requried value. Putting in (1) we have $x + 2 y - 5 = 0$ is the other line.

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3x2−8xy−3y2+10x+20y−25=0 are the bisectors of angle between two lines l1 and l2 one of which passes through origin. Determine the equation of the other line.

$3 x^{2} - 8 x y - 3 y^{2} + 10 x + 20 y - 25 = 0$ are the bisectors of angle between two lines $l_{1}$ and $l_{2}$ one of which passes through origin. Determine the equation of the other line.