Guys i am struggling so much, two questions

1. the directrix of a parabola is x = 0. its focus is (2,6) .
what is the standard form of the parabola?
2. the vertex of a parabola is (−5,2) , and its focus is (−5,6) .
what is the standard form of the parabola?

the equation of the standard parabola

1)

The equation of the standard parabola

2) (x+5)^2 = 16(y-2)

Step-by-step explanation:

Explanation

Parabola:-

The set of points in a plane whose distance from a fixed point and a constant ratio to their corresponding perpendicular distance from a fixed straight line is a conic.

Let S be a fixed point and l be a fixed straight line from any point P,the perpendicular PM is drawn to the line 'l'

1) Step 1 :-

Given the focus   S = (2,6) and directrix is x=0

we know that

now cross multiplication , we get

squaring on both sides,we get

step 2:-

now using distance formula is

Given S =(2,6) and P(x,y) be any point on parabola

........(1)

Now using perpendicular distance formula

let P(x , y ) be any point on the parabola

Given the directrix is x =0 and P(x,y) be any point on parabola

......(2)

equating equation(1) and equation (2), on simplification

we get .....(3)

now the equation (3) is

now the standard form of parabola is

Final answer:-

2) Explanation:-

step 1:

Given vertex of a parabola is A(-5,2) and its focus is S(-5,6)

here the given points of 'x'co- ordinates are equal

now the standard equation of parabola

now you have to find' a' value

Given vertex of a parabola is A(-5,2) and its focus is S(-5,6)

The distance of

 on simplification we get a =4

Final answer :-

the vertex (h,k) = (-5,2) and a=4

The standard parabola is

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

answer: y = -2.5x - 15

              (-6,0)

step-by-step explanation: