Find the equation of the line that is perpendicular to \(2y + 5x - 6 = 0\) and bisects the line joining the points P(4, 3) and Q(-6, 1)

A.

y + 5x + 3 = 0

B.

2y - 5x - 9 = 0

C.

5y + 2x - 8 = 0

D.

5y - 2x - 12 = 0

Correct answer is D

\(Line: 2y + 5x - 6 = 0\)

\(2y = 6 - 5x  \implies y = 3 - \frac{5x}{2}\)

Gradient = \(\frac{-5}{2}\)

For the line perpendicular to the given line, Gradient = \(\frac{-1}{\frac{-5}{2}} = \frac{2}{5}\)

The midpoint of P(4, 3) and Q(-6, 1) = \((\frac{-6 + 4}{2}, \frac{3 + 1}{2})\)

= (-1, 2).

Therefore, the line = \(\frac{y - 2}{x + 1} = \frac{2}{5}\)

\(2(x + 1) = 5(y - 2) \implies 5y - 2x - 12 = 0\)