Correct Answer – Option 3 : 2x – 3y – 8 = 0

Concept:

• The co-ordinates of a point P, dividing the line joining the points A(x₁, y₁) and B(x₂, y₂) in the ratio m : n internally, are given by:

  \(\rm P\left(\dfrac{nx_1+mx_2}{m+n},\dfrac{ny_1+my_2}{m+n} \right)\)

• The equation of a line parallel to the line ax + by + c = 0 is k(ax + by) + c = 0, where k is any non-zero number.

Calculation:

The mid-point divides a line in the ratio 1 : 1 internally.

∴ The co-ordinates of the midpoint (M) of points (1, 3) and (1, -7) will be: \(\rm M\left(\dfrac{1\times1+1\times1}{1+1},\dfrac{1\times3+1\times(-7)}{1+1} \right)\) = M (1, -2).

The equation of the line parallel to the line 2x – 3y – 7 = 0 can be assumed to be k(2x – 3y) – 7 = 0.

Since this line passes through M(1, -2), we will get:

k[2(1) – 3(-2)] – 7 = 0

⇒ k(2 + 6) – 7 = 0

⇒ k = \(\dfrac78\).

The equation, therefore, is:

k(2x – 3y) – 7 = 0

⇒ \(\rm\dfrac78(2x-3y)-7=0\)

⇒ 2x – 3y – 8 = 0.