Find the equation of the line through the intersection of the lines 2x – 3y + 1 = 0 and x + y – 2 = 0 and drawn parallel to y-axis.
Suppose the given two lines intersect at a point P(x1, y1). Then, (x1, y1) satisfies each of the given equations.
2x – 3y + 1 = 0 …(i)
x + y – 2 = 0 …(ii)
Now, we find the point of intersection of eq. (i) and (ii)
Multiply the eq. (ii) by 2, we get
2x + 2y – 4 = 0 …(iii)
On subtracting eq. (iii) from (i), we get
2x – 3y + 1 – 2x – 2y + 4 = 0
⇒ -5y + 5 = 0
⇒ -5y = -5
⇒ y = 1
Putting the value of y in eq. (ii), we get
x + 1 – 2 = 0
⇒ x – 1 = 0
⇒ x = 1
Hence, the point of intersection P(x1, y1) is (1, 1)
The equation of a line parallel to y – axis is of the form
x = a where a is some constant
Given that this equation of the line passing through the point of intersection (1, 1)
Hence, point (1, 1) will satisfy the equation of a line.
Putting x = 1 in the equation y = b, we get
x = a
⇒ 1 = a
or a = 1
Now, required equation of line is x = 1