NIMCET 2016 — Mathematics PYQ

Calculation: Given: focus is (-1, 1) and directrix is 4x + 3y - 24 = 0

$$\Rightarrow y=\frac{-4x}{3}+8$$
So, Slope of directrix = $\frac{-4}{3}$
As we know, Axis of parabola and directrix are perpendicular to each others
So, Slope of directrix × slope of axis = -1
$$\Rightarrow \frac{-4}{3}\times \text{slope of axis}=-1$$
$$\Rightarrow \text{slope of axis}=\frac{3}{4}$$
Now, Equation of axis is written as (y - 1) = $\frac{3}{4}$(x + 1) {Focus lies on axis of parabola}
$$\Rightarrow 4y-4=3x+3$$
$$\Rightarrow 3x-4y+7=0$$
$$\Rightarrow 3x-4y+7=0$$...... (2)
Solving equation (1) and (2), we get
M = (3, 4)
$$\to 4x+3y-24=0$$

The midpoint of M and S is the vertex.
$$\therefore \text{Coordinate of vertex}=\left(\frac{3-1}{2},\frac{4+1}{2}\right)=\left(1,\frac{5}{2}\right)$$