Step-by-step Solution:

Finding the Vertex of the Parabola Given Data: - Focus: \( (-1,1) \) - Directrix: \( 4x + 3y - 24 = 0 \) #Step 1: Find the Midpoint of the Perpendicular from the Focus to the Directrix The midpoint \( M \) of the perpendicular from the focus to the directrix is given by solving the equations of the directrix and the axis of the parabola. Equation of the Axis The axis of the parabola is perpendicular to the directrix. Rewriting the directrix in slope-intercept form: \[ 3y = -4x + 24 \] \[ y = \frac{-4}{3}x + 8 \] The slope of the directrix is \( -\frac{4}{3} \). Since the axis of the parabola is perpendicular to the directrix, we use the property: \[ \text{slope of directrix} \times \text{slope of axis} = -1 \] \[ \left(-\frac{4}{3}\right) \times \text{slope of axis} = -1 \] \[ \text{slope of axis} = \frac{3}{4} \] Using the point-slope form, the equation of the axis, which passes through the focus \( (-1,1) \), is: \[ y - 1 = \frac{3}{4} (x + 1) \] \[ 4(y - 1) = 3(x + 1) \] \[ 4y - 4 = 3x + 3 \] \[ 3x - 4y + 7 = 0 \quad \cdots (2) \] Finding the Midpoint \( M \) Solving the directrix equation and the axis equation: \[ 4x + 3y - 24 = 0 \] \[ 3x - 4y + 7 = 0 \] Multiply the first equation by 4 and the second by 3: \[ 16x + 12y - 96 = 0 \] \[ 9x - 12y + 21 = 0 \] Adding both equations: \[ (16x + 9x) + (12y - 12y) + (-96 + 21) = 0 \] \[ 25x - 75 = 0 \] \[ x = 3 \] Substituting \( x = 3 \) into equation (1): \[ 4(3) + 3y - 24 = 0 \] \[ 12 + 3y - 24 = 0 \] \[ 3y = 12 \] \[ y = 4 \] Thus, the midpoint \( M \) is \( (3,4) \). --- # Step 2: Find the Vertex The vertex is the midpoint of the focus and midpoint \( M \): \[ V = \left( \frac{-1 + 3}{2}, \frac{1 + 4}{2} \right) \] \[ V = \left( \frac{2}{2}, \frac{5}{2} \right) \] \[ V = (1, 5/2) \] Thus, the vertex of the parabola is \( (1, \frac{5}{2}) \). --- Final Answer: \[ \mathbf{(1, 5/2)} \]