Find the equation of the straight line that (i)makes equal intercepts on the axes and passes through the point (2;3) (ii) passes through the point (-5;4) and is such that the portion intercepted between the axes is devided by the point in the ratio `1:2`

Explanation

Let the equation of the line be
`(x)/(a) + (y)/(b) = 1`
Since it makes equal intercepts on the coordinates axes, we have a = b
So, the equation of the line is
`(x)/(a) + (y)/(a) = 1 or x+y =a`
The line passes through the point (2,3). Therefore,
2+3=a
or a = 5
Thus, the equation of the required line is x+y = 5.
(ii) Let the equation of the line be
`(x)/(a) + (y)/(b)= 1`
Clearly, this line meets the coordinate axes at A(a,0) and B(0,b), respectively.
The coordinates of the point that divides the line joining A(a,0) and B(0,b) in the ratio 1:2 are
`((1(0) + 2(a))/(1+2) , (1(b) + 2(0))/(1+2)) -= ((2a)/(3),(b)/(3))`
It is given that the point (-5,4) divides AB in the ratio 1:2.
Therefore, 2a/3 = -5 and b/3 = 4, i.e.,
` a = -(15)/(20) and b= 12`
Hence, the equation of the required line is
`-(x)/(15//2) + (y)/(12) = 1`
or 8x-5y+60 = 0