Question

Find the coordinates of the points of intersection of the lines represented by x² − y² − 2x + 1 = 0

Answer 1

Consider x² − y² − 2x + 1 = 0

∴ (x² − 2x + 1) − y² = 0

∴ (x − 1)² − y² = 0

∴ (x − 1 + y)(x − 1 − y) = 0

∴ (x + y − 1)(x − y − 1) = 0

Separate equations of the lines are,

x + y − 1 = 0 and x − y − 1 = 0

To find the point of intersection of the lines, we have to solve

x + y − 1 = 0   ...(1)

and x − y − 1 = 0    ...(2)

Adding equations (1) and (2), we get,

2x − 2 = 0

2x = 2

x = `2/2`

∴ x = 1

Substituting x = 1 in (1), we get,

1 + y − 1 = 0

y = 1 − 1

∴ y = 0

∴ The coordinates of the point of intersection of the lines are (1, 0).

Answer 2

The general second-degree equation representing a pair of straight lines is:

ax² + 2hxy + by² + 2gx + 2fy + c = 0

Comparing it with the given equation:

x² − y² − 2x + 1 = 0

We identify the coefficients:

a = 1, h = 0, b = −1, g = −1, f = 0, c = 1

The condition for the lines to intersect is:

h² − ab ≥ 0 

Substituting the values:

0² − (1)(−1) = 1 ≥ 0

Since the condition holds, the lines do intersect.

The coordinates of the point of intersection are given by:

= `((hf - bg)/(ab - h^2), (gh - af)/(ab - h^2))`

= `((0 xx 0 - (-1) xx (-1))/(1 xx (-1) xx (0)^2), ((-1) xx 0 - 1 xx 0)/(1 xx (-1) - (0)^2))`

= `((-1)/-1, 0/-1)`

= (1, 0)