Question 1 :
Find the equation of a line passing through the point of intersection of the lines 4x + 7y − 3 = 0 and 2x − 3y + 1 = 0 that has equal intercepts on the axes.
Answer :
First let us find the point of intersection of the lines 4x + 7y − 3 = 0 and 2x − 3y + 1 = 0
4x + 7y − 3 = 0 ------(1)
2x − 3y + 1 = 0 ------(2)
3(1) + 7(2)
12x + 21y − 9 = 0
14x − 21y + 7 = 0
---------------------
26x - 2 = 0
x = 2/26
x = 1/13
Let us apply the value of x in (1), we get
4(1/13) + 7y - 3 = 0
7y - 3 + (4/13) = 0
7y = 3 - (4/13)
7y = (39-4)/13
7y = 35/13
y = (35/13 x 7)
y = 5/13
So, the point of intersection of the given lines is (1/13, 5/13). Since the required line is having equal intercepts, a = b
(x/a) + (y/b) = 1
The line is passing through the point (1/13, 5/13)
((1/13)/a) + ((5/13)/a) = 1
(1/13a) + (5/13a) = 1
6/13a = 1
13a = 6
a = 6/13
Equation of the line :
(x/(6/13)) + (y/(6/13)) = 1
(13x + 13y)/6 = 1
13x + 13y = 6
13x + 13y - 6 = 0
Question 2 :
A person standing at a junction (crossing) of two straight paths represented by the equations 2x −3y + 4 = 0 and 3x + 4y −5 = 0 seek to reach the path whose equation is 6x −7y + 8 = 0 in the least time. Find the equation of the path that he should follow.
Answer :
The equation of the given lines are
2x −3y + 4 = 0 -----(1)
3x + 4y −5 = 0 -----(2)
6x −7y + 8 = 0 -----(3)
the person is standing at the junction of paths represented by the line (1) and (2)
By solving the (1) and (2), we get
4(1) + 3(2)
8x - 12y + 16 = 0
9x + 12y - 15 = 0
--------------------
17x + 1 = 0
x = -1/17
By applying the value of x in (1), we get
2(-1/17) - 3y + 4 = 0
-3y = -4 + (2/17)
-3y = (-68 + 2)/17
-3y = - 66/17
y = 66/17(3) = 22/17
Thus the person is standing at the point (-1/17, 22/17)
The person can reach path (3) is the least time if he takes along the perpendicular the line to (3) from the point (-1/17, 22/17)
Slope of the line (3)
m = - coefficient of x/coefficient of y
m = -6/7
Equation of the line passing through the point (-1/17, 22/17) and having the slope -6/7
Slope of the required line = 7/6
(y - y_{1}) = m(x - x_{1})
(y - (22/17)) = (7/6) (x + (1/17))
(17y - 22)/17 = (7/6)(17x + 1)/17
6(17y - 22) = 7(17x + 1)
102y - 132 = 119x + 7
119x + 102y + 132 - 7 = 0
119 x + 102y + 125 = 0
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