If the focus of the parabola X & # 178; = 2PY (P > 0) coincides with the upper focus of the ellipse X & # 178 / / 3 + Y & # 178 / / 4 = 1, 1) solve the parabolic equation 2) if AB is For the moving chord passing through the focus of the parabola, the straight lines L1 and L2 are the two tangent lines of the parabola tangent to a and B respectively, and the ordinates of the focus of L1 and L2 are calculated

If the focus of the parabola X & # 178; = 2PY (P > 0) coincides with the upper focus of the ellipse X & # 178 / / 3 + Y & # 178 / / 4 = 1, 1) solve the parabolic equation 2) if AB is For the moving chord passing through the focus of the parabola, the straight lines L1 and L2 are the two tangent lines of the parabola tangent to a and B respectively, and the ordinates of the focus of L1 and L2 are calculated

（1）x²=4y
（2）y=-2

Given the distance √ 2 / 2 between the point on the parabola X & # 178; = 2PY (P > 0) and the straight line lx-y-2, the standard equation of parabola is obtained
Given that the distance between the point on the parabola X & # 178; = 2PY (P > 0) and the line lx-y-2 is the nearest √ 2 / 2, the standard equation of parabola is obtained

Let the point on the parabola nearest to the line l be q (x0, Y0), then the tangent passing through q is parallel to the line L
Let the tangent passing through Q be x0x = P (y + Y0), that is, x0x-py-py0 = 0
Then x0 = P (I)
And the distance from Q to line L is
|x0-y0-2|/√2=√2/2（II）
And point q is on a parabola
x0^2=2py0（III）
(P / 2-2) ^ 2 = 1 from (I) (II) (III)
The solution is p = 2 or P = 6
So the parabolic equation is
X ^ 2 = 4Y or
x^2=12y

Find the shortest distance from the point on the parabola y = x ^ 2 to the line x-y-2 = 0
Thanks for the answer!

Let X-Y + B = 0 be a tangent equation of parabola parallel to x-y-2 = 0
The simultaneous equations are: x ^ 2-x-b = 0
△=1+4b=0 b=-1/4
It is easy to know that the shortest distance between a point on a parabola and a straight line x-y-2 = 0 is the distance between two parallel straight lines
dmin=|2-(-1/4)|/√2=(9√2)/8
Glad to solve the problem for you!