Correct Option: A To determine which of the given points lies on the line $L$ passing through $A(3, 4)$ and $B(7, -2)$, we first need to find the equation of the line $L$. **Step 1: Calculate the slope of the line $L$.** The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Given points are $A(3, 4)$ and $B(7, -2)$. Let $(x_1, y_1) = (3, 4)$ and $(x_2, y_2) = (7, -2)$. $m = \frac{-2 - 4}{7 - 3} = \frac{-6}{4} = -\frac{3}{2}$ **Step 2: Determine the equation of the line $L$.** We can use the point-slope form of the equation of a line, $y - y_1 = m(x - x_1)$. Using point $A(3, 4)$ and the slope $m = -\frac{3}{2}$: $y - 4 = -\frac{3}{2}(x - 3)$ Multiply both sides by 2 to clear the fraction: $2(y - 4) = -3(x - 3)$ $2y - 8 = -3x + 9$ Rearrange the terms to get the standard form of the equation $Ax + By = C$: $3x + 2y = 9 + 8$ $3x + 2y = 17$ This is the equation of line $L$. Any point $(x, y)$ that lies on line $L$ must satisfy this equation. **Step 3: Check each option by substituting its coordinates into the equation $3x + 2y = 17$.** A) Point $(5, 1)$: Substitute $x=5$ and $y=1$ into the equation: $3(5) + 2(1) = 15 + 2 = 17$ Since $17 = 17$, the point $(5, 1)$ satisfies the equation. Thus, it lies on line $L$. B) Point $(1, 8)$: Substitute $x=1$ and $y=8$ into the equation: $3(1) + 2(8) = 3 + 16 = 19$ Since $19 \neq 17$, the point $(1, 8)$ does not lie on line $L$. C) Point $(-1, 9)$: Substitute $x=-1$ and $y=9$ into the equation: $3(-1) + 2(9) = -3 + 18 = 15$ Since $15 \neq 17$, the point $(-1, 9)$ does not lie on line $L$. D) Point $(0, 7)$: Substitute $x=0$ and $y=7$ into the equation: $3(0) + 2(7) = 0 + 14 = 14$ Since $14 \neq 17$, the point $(0, 7)$ does not lie on line $L$. Therefore, only option A satisfies the equation of the line.