Consider the family of lines #(4a+3)x - (a+1)y - (2a+1) = 0# where #ainR#. Minimum area of the triangle which a member of this family with negative gradient can make with the positive semi axes?
A) 8
B) 6
C) 4
D) 2
A) 8
B) 6
C) 4
D) 2
1 Answer
Option (C)
Explanation:
Given family of lines
Rearranging we get
Obviously the all lines of the family mus pass through the point of intersection of two lines represented by the following two equations
and
#3x-y-1=0........[2]
Subtracting [2] from [1] we get
Inserting
So the coordinates of common point of intersection of all lines of the family will be
Any line passing through this common point
Now rearranging [3] in intercept form we get
So area of the triangle made by this line with the positive semi axes will be given by
Differentiating w r, to m we get
Imposing the condition of minimization of
Hence minimum area of the the triangle should be for m=-2##
[It matches with Option (C)]