Question

If the focus of a parabola is (-2,1) and the directrix has the equation x+y=3,then its vertex is

• A. (0,3)
• B. $-1,\frac{1}{2}$
• C. (-1,2)
• D. (2,-1)

Answer 1

The correct option is C

(-1,2)

Given:The focus S is at (-2,1) and the directrix is the line x+y-3=0

The slope of the line perpendicular to x+y-3=0 is 1.

The axis of the parabola is perpendicular to the directrix and passes through the focus

$\therefore \text{Equation of the axis of the parabola=y-1=x(1+2)}..\left(1\right)$

Intersection point of the dirctrix and the axis is the intersection point of (1) and x+y-3=0

Let the intersection point be K

Therefore,the coordinates of K will be (0,3)

Let (h,k) be the coordinates of the vertex which is the mid-point of the segment joining K and the focus.

$\therefore h=\frac{0-2}{2},k=\frac{3+1}{2}$

$h=-1,k=2$

Hence,the coordinates of the vertex are (-1,2)