Allow us to consider the co-ordinates of the foot of the opposite from (- 1, 3) to the line \[\mathbf{3x}\text{ }\text{ }\mathbf{4y}\text{ }\text{ }\mathbf{16}\text{ }=\text{ }\mathbf{0}\text{ }\mathbf{be}\text{ }\left( \mathbf{a},\text{ }\mathbf{b} \right)\]

Thus, let the incline of the line joining (- 1, 3) and (a, b) be m1

\[\mathbf{m1}\text{ }=\text{ }\left( \mathbf{b}-\mathbf{3} \right)/\left( \mathbf{a}+\mathbf{1} \right)\]

Also, let the incline of the line \[\mathbf{3x}\text{ }\text{ }\mathbf{4y}\text{ }\text{ }\mathbf{16}\text{ }=\text{ }\mathbf{0}\] be m2

\[\mathbf{y}\text{ }=\text{ }\mathbf{3}/\mathbf{4x}\text{ }\text{ }\mathbf{4}\]

\[\mathbf{m2}\text{ }=\text{ }\mathbf{3}/\mathbf{4}\]

Since these two lines are opposite, \[\mathbf{m1}\text{ }\times \text{ }\mathbf{m2}\text{ }=\text{ }-\text{ }\mathbf{1}\]

\[\left( \mathbf{b}-\mathbf{3} \right)/\left( \mathbf{a}+\mathbf{1} \right)\text{ }\times \text{ }\left( \mathbf{3}/\mathbf{4} \right)\text{ }=\text{ }-\text{ }\mathbf{1}\]

\[\left( \mathbf{3b}-\mathbf{9} \right)/\left( \mathbf{4a}+\mathbf{4} \right)\text{ }=\text{ }-\text{ }\mathbf{1}\]

\[\mathbf{3b}\text{ }\text{ }\mathbf{9}\text{ }=\text{ }-\text{ }\mathbf{4a}\text{ }\text{ }\mathbf{4}\]

\[\mathbf{4a}\text{ }+\text{ }\mathbf{3b}\text{ }=\text{ }\mathbf{5}\text{ }\ldots \text{ }.\left( \mathbf{1} \right)\]

Point (a, b) lies on the line \[\mathbf{3x}\text{ }\text{ }\mathbf{4y}\text{ }=\text{ }\mathbf{16}\]

\[\mathbf{3a}\text{ }\text{ }\mathbf{4b}\text{ }=\text{ }\mathbf{16}\text{ }\ldots \text{ }..\left( \mathbf{2} \right)\]

Tackling conditions (1) and (2), we get

\[\mathbf{a}\text{ }=\text{ }\mathbf{68}/\mathbf{25}\text{ }\mathbf{and}\text{ }\mathbf{b}\text{ }=\text{ }-\text{ }\mathbf{49}/\mathbf{25}\]

∴ The co-ordinates of the foot of opposite is \[\left( \mathbf{68}/\mathbf{25},\text{ }-\text{ }\mathbf{49}/\mathbf{25} \right)\]