The equation of the line through the intersection of the lines `2x-3y=0` and `4x-5y=2` and

Answer

N/a

Explanation

Given equation of the lines are
`2x-3y=0`
and `4x-5y=2`
From Eq. (i), put `x=(3y)/(2)` in Eq. (ii), we get
`4.(3y)/(2)-5y=2`
`rArr 6y-5y=2`
`rArr y=2`
Now, put `y=2` in Eq. (i), we get
x=3
So, the intersection points are (3,2).
(i) The equation of the line passes through the point (3,2) and (2,1), is
`y-2=(1-2)/(2-3)(x-3)`
`rArry-2=(x-3)`
`rArr x-y-1=0`
(ii) If the required line is perpendicular to the line `x+2y+1=0`
`:'` Slope of the required line =2
`:.` Equation of the line is
`y-2=2(x-3)`
`rArr 2x-y-4=0`
(iii) If the required line is parallel to the line `3x-4y+5=0`, then the slope of the required line `=(3)/(4)`
`:.` Equation of the required line is
`y-2=(3)/(4)(x-3)`
`rArr 4y-8=3x-9`
`rArr 3x-4y-1=0`
(iv) If the line is equally inclined to the X-axis, then
`m=+-tan45^(@)=+-1`
`:.` Equation of the line is
`y-2=-1(x-3)" "`[taking negative value]
`rArr y-2=-x+3`
`rArr x+y-5=0`
So, the correct matches are (a) `to` (iii), (b) `to` (i), (c) `to` (iv), (d) `to` (ii)