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Question
Find the equation of the 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.
Solution
4x + 7y = 3 ..…(i)
2x – 3y = –1 .....(ii)
On multiplying equation (ii) by 2,
4x – 6y = – 2 ..…(iii)
By subtracting equation (iii) from (i)
13y = 5
∴ y = `5/13`
Putting the value of y in equation (i),
`4"x" + 7 xx 5/13 = 3`
or `4"x" = 3 - 35/13`
= `(39 - 35)/13`
= `4/13`
∴ `"x" = 1/13`
The given lines intersect at the point `(1/3, 5/13)`.
Lines whose intercepts on the axes are equal make an angle of 45° or 135° with the positive x-axis. Therefore its slope will be ±1.
∴ Equations of lines PA and PB
y – y1 = m(x – x1)
(i) When m = 1, then y – `5/13 = 1 ("x" - 1/13)`
or 13y – 5 = 13 x – 1 or 13x – 13y + 4 = 0
(ii) When m = –1, then y – `5/13 = 1 ("x" - 1/3)`
13y – 5 = –13x + 1
∴ 13x + 13y – 6 = 0
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