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 |
---|---|---|

+ | use + sign in eq.(1) | use – sign in eq.(1) |

– | use – sign in eq.(1) | use + sign in eq.(1) |

## The combined equation of angle bisectors :

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.

**Note :**

(i) If a = b, then bisectors are \(x^2-y^2\) = 0 i.e. x – y = 0, x + y = 0

(ii) If h = 0, the bisectors are xy = 0 i.e. x = 0, y = 0

(iii) The two bisectors are always at right angles, since we have coefficient of \(x^2\) + coefficient of \(y^2\) = 0