Question

the focus and directrix of a parabola are (1,-1)and x+y+3=0 its vertex is​

Answer 1

Step-by-step explanation:

We know that, vertex of a parabola lies on the perpendicular distance in between focus and directrix and lies on the axis of parabola

Now,

Equation of directrix x + y + 3 = 0 .....(i)

slope of directrix, M= -1

So, slope of axis, m= 1

Equation of axis given by,

y = mx + c

=> y = x + c .... (ii)

The point (1, -1) lies on the line (ii),

so,

-1 = 1 + c

=> c = -2

So, equation of axis is

x - y - 2 = 0 (iii)

Adding and subtracting (i) and (iii),

$x = \frac{ - 1}{2} \: \: and \: \: y = \frac{ - 5}{2}$

Now, the vertex of the parabola will be the midpoint of the points (1, -1) and (-1/2 , -5/2)

So, the required coordinates of vertex are

$x = \frac{1}{4} \: \: and \: \: y = \frac{ - 7}{4}$

the coordinates are (1/4 , -7/4 )