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Question
Answer the following question:
Show that there are two lines which pass through A(3, 4) and the sum of whose intercepts is zero.
Solution
Case I: Line not passing through origin.
Let the equation of the line be `x/"a" + y/"b"` = 1 ...(i)
This line passes through (3, 4)
∴ `3/"a" + 4/"b"` = 1 ...(ii)
Since the sum of the intercepts of the line is zero,
a + b = 0
∴ a = – b ...(iii)
Substituting the value of a in (ii), we get
`3/(-"b") + 4/"b"` = 1
∴ `1/"b"` = 1
∴ b = 1
∴ a = – 1 ...[From (iii)]
Substituting the values of a and b in (i), the equation of the required line is
`x/(-1) + y/1` = 1
∴ x – y = – 1
∴ x – y + 1 = 0
Case II: Line passing through origin.
Slope of line passing through origin and A(3, 4) is
m = `(4 - 0)/(3 - 0) = 4/3`
∴ Equation of the line having slope m and passing through origin (0, 0) is y = mx.
∴ The equation of the required line is
y = `4/3x`
∴ 4x – 3y = 0
∴ There are two lines which pass through A(3, 4) and the sum of whose intercepts is zero.
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