Question 19

Explanation

Given line $g_1$ passes through point $A(-1, 1)$ and $B(2, 3)$, and the equation of line $g_2$: $mx + ny - 7 = 0$.

Calculating the Gradient of Line g₁

The gradient of a line passing through two points can be calculated as:

For line $g_1$ passing through $A(-1, 1)$ and $B(2, 3)$:

Therefore, the gradient of line $g_1$ is $m_1 = \frac{2}{3}$.

Determining the Gradient of Line g₂

The equation of line $g_2$: $mx + ny - 7 = 0$.

Convert to the form $y = mx + c$:

Therefore, the gradient of line $g_2$ is $m_2 = -\frac{m}{n}$.

Evaluating Statement (1)

Statement: Line $g_1$ and $g_2$ are parallel if $m = 4$ and $n = 6$.

Two lines are parallel if their gradients are equal.

If $m = 4$ and $n = 6$, then:

Since $m_1 = \frac{2}{3}$ and $m_2 = -\frac{2}{3}$, then $m_1 \neq m_2$.

Therefore, statement $(1)$ is false.

Evaluating Statement (2)

Statement: Point $B$ passes through line $g_2$ if $m = 2$ and $n = -1$.

Point $B(2, 3)$ passes through line $g_2$ if it satisfies the equation $mx + ny - 7 = 0$.

Substitute $m = 2$, $n = -1$, and $B(2, 3)$:

Since the result is not equal to $0$, point $B$ does not pass through line $g_2$.

Therefore, statement $(2)$ is false.

Evaluating Statement (3)

Statement: If $g_1$ and $g_2$ are parallel, then $\frac{m}{n} = -\frac{3}{2}$.

If both lines are parallel, then $m_1 = m_2$:

Therefore, if $g_1$ and $g_2$ are parallel, then $\frac{m}{n} = -\frac{2}{3}$, not $-\frac{3}{2}$.

Therefore, statement $(3)$ is false.

Evaluating Statement (4)

Statement: If $m = 3$ and $n = 2$, then $g_1$ and $g_2$ are perpendicular.

Two lines are perpendicular if the product of their gradients equals $-1$, i.e., $m_1 \times m_2 = -1$.

If $m = 3$ and $n = 2$, then:

Check if $m_1 \times m_2 = -1$:

Since $m_1 \times m_2 = -1$, then $g_1$ and $g_2$ are perpendicular.

Therefore, statement $(4)$ is true.