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Question
For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.
Solution
The given equation are ax + by + 8 = 0 ......(i)
And 2x – 3y + 6 = 0 ......(ii)
From equation (i) we get,
ax + by + 8 = 0
⇒ `a/(-8)x + b/(-8)y` = 1
⇒ `x/((-8)/a) + y/((-8)/b)` = 1
So, the intercepts on the axes are `(-8)/a` and `(-8)/b`
From equation (ii), we get
2x – 3y + 6 = 0
⇒ 2x – 3y = – 6
⇒ `(2x)/(6) - (3y)/(-6)` = 1
⇒ `x/(-3) + y/2` = 1
So, the intercepts are – 3 and 2.
`(-8)/a` = + 3
⇒ a = ` - 8/3`
⇒ `(-8)/b` = – 2
⇒ b = + 4
Hence, the required values of a and b are `(-8)/3` and 4.
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