# Family of Lines – The combined Equation of Angle Bisectors

Here, you will learn equation of family of lines and the combined equation of the bisectors of the angle between the lines.

## Family of Lines Equation

If equation of two lines be P = $$a_1x+b_1y+c_1$$ = 0 and Q = $$a_2x+b_2y+c_2$$ = 0, then the equation of the lines passing through the point of intersection of these lines is : P + $$\lambda$$Q = 0 or $$a_1x+b_1y+c_1$$ + $$\lambda$$$$a_2x+b_2y+c_2$$ = 0. The value of $$\lambda$$ is obtained with the help of the additional information given in the problem.

Example : Prove that each member of the family of straight lines ($$3sin\theta + 4cos\theta$$)x + ($$2sin\theta – 7cos\theta$$)y + ($$sin\theta + 2cos\theta$$) = 0 ($$\theta$$ is a parameter) passes through a fixed point.

Solution : The given family of straight lines can be rewritten as

(3x+2y+1)$$sin\theta$$ + (4x-7y+2)$$cos\theta$$ = 0

or, (4x-7y+2) + $$tan\theta$$(3x+2y+1) = 0 which is of the form $$L_1$$ + $$\lambda L_2$$ = 0

Hence each member of it will pass through a fixed point which is the intersection of 4x-7y+2 = 0 and 3x+2y+1 = 0

i.e. ($$-11\over 29$$,$$2\over 29$$)

## Equation of bisectors of angles between two lines :

If equation of two intersecting lines are $$a_1x+b_1y+c_1$$ = 0 and $$a_2x+b_2y+c_2$$ = 0, then equation of bisectors of the angles between these lines are written as :

$$a_1x+b_1y+c_1\over {\sqrt{{a_1}^2 + {b_1}^2}}$$ = $$\pm$$ $$a_2x+b_2y+c_2\over {\sqrt{{a_2}^2 + {b_2}^2}}$$   …..(i)

(a)  Equation of bisector of angle containing origin :  If the equation of the lines are written with constant terms $$c_1$$ and $$c_2$$ positive, then the equation of the bisectors of the angle containing the origin is obtained by taking positive sign in (i)

(b)  Equation of bisector of acute/obtuse angle : To find the equation of the bisector of the acute or obtuse angle :

(i)  Let $$\phi$$ be angle between one of the two bisectors and one of the two given lines. Then if tan$$\phi$$ < 1

i.e. $$\phi$$ < 45 i.e. 2$$\phi$$ < 90, the angle bisector will be bisector of acute angle.

(ii)  See whether the constant terms $$c_1$$ and $$c_2$$ in the two equation are +ve or not. If not then multiply both sides of given equation by -1 to make the constant terms positive.

Determine the sign of  $$a_1a_2+b_1b_2$$

If sign of $$a_1a_2+b_1b_2$$ for obtuse angle bisector for acute angle bisector
The combined equation of angle bisectors between the lines represented by homogeneous equation of second degree is given by $$x^2-y^2\over {a-b}$$ = $$xy\over h$$, a $$\ne$$ b, h $$\ne$$ 0.
(i)  If a = b, then bisectors are $$x^2-y^2$$ = 0 i.e. x – y = 0, x + y = 0
(iii)  The two bisectors are always at right angles, since we have coefficient of $$x^2$$ + coefficient of $$y^2$$ = 0