Question

Find equations of lines which contains the point A(1, 3) and the sum of whose intercepts on the coordinate axes is zero.

Answer 1

Let the intercepts made by the line on the coordinate axes be a and b respectively.

∴ a + b = 0   ...(1)

The equation of the line is `x/"a" + y/"b"` = 1.

Since the line passes through the point A(1, 3),

`1/"a" + 3/"b"` = 1

∴ b + 3a = ab  ...(2)

From (1), a = – b

Substituting a = – b in (2), we get,

b – 3b = – b(b)

∴ – 2b = – b²

∴ b² – 2b = 0

∴ b(b – 2) = 0

∴ b = 0 or b – 2 = 0

∴ b = 0 or b = 2

By (1), when b = 0, a = 0

and when b = 2, a = – 2

When a = – 2, b = 2, equation of the line is

`x/(-2) + y/2` = 1

∴ – x + y = 2

∴ x – y + 2 = 0

When a = 0, b = 0, the line is passing through the origin.

∴ its equation is

y = mx

Since this line passes through A(1, 3),

3 = m(1)

∴ m = 3

∴ equation of the line is

y = 3x, i.e., 3x – y = 0

Hence, equations of required lines are x – y + 2 = 0 and 3x – y = 0.