What is the standard form equation of the parabola with a directrix of x=5 and focus at (11, -7)?
2 Answers
Standard form is:
Explanation:
Because the directrix is a vertical line,
where (h,k) is the vertex and #f is the signed horizontal distance from the vertex to the focus.
We know that the y coordinate, k, of the vertex is the same as the y coordinate of the focus:
Substitute -7 for k into equation [1]:
We know that the x coordinate of the vertex is the midpoint between the x coordinate of the focus and the x coordinate of the directrix:
Substitute 8 for h into equation [2]:
The focal distance is the signed horizontal distance from the vertex to the focus:
Substitute 3 for f into equation [3]:
We will multiply the denominator and write -- as +
Expand the square:
Distribute the
Combine the constant terms:
#x=y^2/12+7/6y+145/12#
Explanation:
Directrix
Focus
From this we can findout the vertex.
Look at the diagram
Vertex lies exactly in between Directrix and Focus
#x,y=(5+11)/2, (-7 + (-7))/2=(8, -7)#
The distance between Focus and vertex is
The parabola is opening to the right
The equation of the Parabola here is -
#(y-k)^2=4a(x-h)#
#(h,k)# is the vertex
#h=8#
#k=-7#
Plugin
#(y-(-7))^2=4.3(x-8)#
#(y+7)^2=4.3(x-8)#
#12x-96=y^2+14y+49# [by transpose]
#12x=y^2+14y+49+96#
#12x=y^2+14y+145#
#x=y^2/12+14/12y+145/12#
#x=y^2/12+7/6y+145/12#