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Basic Concepts Of Straight line Part 4

Posted on - 04-01-2017

Math

IIT JEE

Family of Lines

The general equation of the family of lines through the point of intersection of two given lines is L + lL¢ = 0, where L = 0 and L¢ = 0 are the two given lines, and l is a parameter. Conversely, any line of the form L1 + l L2 = 0 passes through a fixed point which is the point of intersection of the lines L1 = 0 and L2 = 0.

  • The family of lines perpendicular to a given line ax + by + c = 0 is given by
    bx – ay + k = 0, where k is a parameter.
  • The family of lines parallel to a given line ax + by + c = 0 is given by
    ax + by + k = 0, where k is a parameter.

    Example 1

    A variable line through the point of intersection of the lines x/a + y/b=1 and x/b + y/a = 1 meets the coordinate axes in A and B. Show that the locus of the midpoint of AB is the curve: 2xy(a + b) = ab(x + y).

    Solution:

    Let (h, k) be the mid point of the variable line AB. .

    The equation of the variable line AB is

    (bx + ay – ab) + l(ax + by - ab) = 0

    Coordinates of A are

    Coordinate of B are

    Mid point of AB is


    &⇒
    h = k =


    &⇒


    &⇒

    &⇒
    (h + k)ab = 2hk (a + b).

    Hence the locus of the mid-point of AB is

    (x + y) ab = 2xy (a + b).

    One Parameter Family of Straight Lines:

    If a linear expression L1 contains an unknown coefficient, then the line L1 =0 can not be a fixed line. Rather it represents a family of straight lines known as one parameter family of straight lines. e.g. family of lines parallel to the x-axis i.e. y = c and family of straight lines passing through the origin i.e. y = mx.

    Each member of the family passes through a fixed point. We have two methods to find the fixed point. .

    Method (i):

    Let the family of straight lines be of the form ax + by + c = 0 where a, b, c are variable parameters satisfying the condition al + bm + cn = 0, where l, m, n are given and n > 0. Rewriting the condition as , and comparing with the given family of straight lines, we find that each member of it passes through the fixed point .

    Example 2

    If a, b and c are three consecutive odd integers then prove that the variable line ax + by + c = 0 always passes through (1, –2).

    Solution:

    Since a, b and c are three consecutive odd integers, these must be in A.P.
    &⇒
    2b = a + c or a - 2b + c = 0
    . .

    Hence, the variable line always passes through (1, –2).

    Example 3

    If the algebraic sum of perpendiculars from n given points on a variable straight line is zero then prove that the variable straight line passes through a fixed point. .

    Solution:

    Let n given points be (xi, yi) where i = 1, 2…. n and the variable straight line is ax + by + c = 0. Given that


    &⇒
    a åxi + båyi + cn = 0
    &⇒
    .

    Hence the variable straight line always passes through the fixed point .

    Method (ii):

    If a family of straight lines can be written as L1 + lL2 = 0 where L1, L2 are two fixed lines and l is a parameter, then each member of it will pass through a fixed point given by point of intersection of L1 = 0 and L2 = 0.

    Note: If L1 = 0 and L2 = 0 are parallel lines , they will meet at infinity.

    Example 4

    Prove that each member of the family of straight lines
    (3sinq + 4cosq)x + (2sinq - 7cosq)y + (sinq + 2cosq) = 0 (q is a parameter) passes through a fixed point.

    Solution:

    The given family of straight lines can be rewritten as

    (3x + 2y + 1) sinq + (4x – 7y + 2) cosq = 0

    or, (4x – 7y + 2) + tanq (3x + 2y + 1) = 0

    which is of the form L1 + l L2 = 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. .

    Concurrency of Straight Lines:

    The condition for 3 lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, a3x + b3y + c3 = 0 to be concurrent is

    (i) .

    (ii) There exist 3 constants l, m, n (not all zero at the same time) such that
    l L1 + mL2 + nL3 = 0, where L1 = 0, L2 = 0 and L3 = 0 are the three given straight lines.

    (iii) The three lines are concurrent if any one of the lines passes through the point of intersection of the other two lines.

     
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