High school mathematics conic problem to detailed process Given that x ^ 2 / 6 - y ^ 2 / 3 of the hyperbola = 1, D points are f1.f2, m points are on the hyperbola, and MF1 is perpendicular to the X axis, what is the distance from F1 to the straight line F2m? Detailed must be detailed process

High school mathematics conic problem to detailed process Given that x ^ 2 / 6 - y ^ 2 / 3 of the hyperbola = 1, D points are f1.f2, m points are on the hyperbola, and MF1 is perpendicular to the X axis, what is the distance from F1 to the straight line F2m? Detailed must be detailed process

M (- 2,3) or (- 2, - 3)
Find MF &; = 5
s△MF₁F₂=6
So the required distance is 5 / 12

The product of the slope of the line between M and F1 (- N, 0), F2 (n, 0) is a constant M. when the locus of M is a hyperbola with eccentricity root 3, M =?

Let m (x, y) know that M = Y / (x-n) × Y / (x + n), that is, X & sup2; / N & sup2; - Y & sup2; / (Mn & sup2;) = 1. From the equation form, F1 and F2 are not the focus of hyperbola, but the vertex. A & sup2; = n & sup2;, B & sup2; = Mn & sup2;, C & sup2; = (M + 1) n & sup2;, have eccentricity = √ 3, and E & sup2; = n & sup2;, C & sup2; = (M + 1) n & sup2;, have eccentricity = √ 3

1. The focus of the parabola C: y square = 2px (P > 0) is f, the line L passing through f intersects the parabola C at P and Q, and the vector PQ = - 2, the vector FQ
(1) The equation for finding the line L
(2) If | PQ | = 9 / 2, find the equation of this parabola
2. It is known that the ratio of the distance from a moving point P in the plane to the root 3 / 3 of the line L: x = 4 and the distance to the fixed point F (root 3,0) is 2 root 3 / 3, and the trajectory of the moving point P is set as C
(1) Finding the equation of trajectory C
(2) Let F1 and F2 be the left and right focus of C respectively, and find the maximum and minimum of | Pf1 | multiplied by | PF2 |
(3) Let the line L passing through the fixed point m (0,2) intersect with the track C at two different points a and B, and the angle AOB is an acute angle (where 0 is the origin of the coordinate), and find the value range of the slope k of the line L

Let p be above the x-axis,
Let L: y = K (X-P / 2),
And Y ^ 2 = 2px, eliminate x, get
y(P)*y(Q)=-p^2
From the question, we can get the conclusion
y(P)=-2*y(Q)
It can be solved by two formulas
y(P)=p*√2,
y(Q)=-p*√2/2.
So the slope of PQ is
k=2p/[y(P)+y(Q)]
=2√2.
So the equation of PQ is:
y=2√2(x-p/2).
According to the symmetry, we get
When p of PQ is below x-axis, the equation is as follows:
y=-2√2(x-p/2).
thank you!

It is proved that if Liman = a, Lim | an | = | a | otherwise, it is not true

It is proved that: ∵ Liman = a
For any ε > 0, there exists n > 0. When n > N, there is an-a-0. When n > N, there is an an-a-0

Find LIM (x → 8) [(9 + 2x) ^ 0.5-5] / [x ^ (1 / 3) - 2]

The limit of LIM [(1-2x) ^ 1 / 3 - (1 + 2x) ^ 1 / 4] / sin3x at x → 0

According to (1 + ax) ^ B-1 ~ ABX in the molecule + 1 and - 1 when X -- > 0, we can get this form

lim(x→0)2+[(x^3-2x^2+1)/(x-1)]
It's best to come up with a solution

If it's meaningful, the limit is this. If it's meaningless, for example, when the denominator is 0, try to divide it, or rationalize the denominator
lim(x→0)2+[(x^3-2x^2+1)/(x-1)]
=2+(1/(-1))
=1.

What is LIM (x - > + ∞) e ^ (- 2x) equal to

-2x->-∞
So LIM (x - > + ∞) e ^ (- 2x) = 0

LIM (1-2 / x) ^ 2x =? X tends to infinity

LIM (1-2 / x) ^ 2x (x tends to infinity)
=Lim [1 + 1 / (- X / 2)] ^ [(- X / 2) * (- 4)] (x tends to infinity)
=Lim {[1 + 1 / (- X / 2)] ^ (- X / 2)} ^ (- 4) (x tends to infinity)
=e^(-4)

LIM (x ^ 2 + 1) / (2x ^ + 1) x tends to 0

Is the denominator 2x ^?
If it's 1, it's 0; if it's 2, it's 1; if it's greater than 2, it's ∞