If P (x, y) is a point on the ellipse x ^ 2 & # 47; 4 + y ^ 2 & # 47; 9 = 1, then z = 2x + y is the maximum If P (x, y) is a point on the ellipse x ^ 2 / 4 + y ^ 2 / 9 = 1, what is the maximum value of Z = 2x + y

If P (x, y) is a point on the ellipse x ^ 2 & # 47; 4 + y ^ 2 & # 47; 9 = 1, then z = 2x + y is the maximum If P (x, y) is a point on the ellipse x ^ 2 / 4 + y ^ 2 / 9 = 1, what is the maximum value of Z = 2x + y

What is the maximum value of Z = 2x + y if P (x, y) is a point on the ellipse X & # 178 / 4 + Y & # 178 / 9 = 1
Let x = 2cost, y = 3sint, then z = 4cost + 3sint = 4 [cost + (3 / 4) Sint]
[let Tan θ = 3 / 4, sin θ = 3 / 5, cos θ = 4 / 5]
=4(cost+tanθsint)=4[cost+(sinθ/cosθ)sint]
=(4/cosθ)(costcosθ+sintsinθ)=(4/cosθ)cos(t-θ)
=5cos[t-arctan(3/4)]
So when t = arctan (3 / 4), the maximum value of Z is 5

It is known that the center of the ellipse C is at the origin of the coordinate, the focus is on the x-axis, the eccentricity is 1 | 2, and the maximum distance between the point on the ellipse C and the focus is 3. The standard equation of the ellipse C
How to use the maximum focus distance?

The maximum distance from the point on the ellipse C to the focus is 3
So a + C = 3
e=c/a=1/2
We get a = 2, C = 1
So the standard equation of ellipse is x ^ 2 / 4 + y ^ 2 / 3 = 1

It is known that the center of the ellipse C is at the origin of the coordinate, the focus is on the x-axis, the eccentricity is 1.2, and the maximum distance from the point on the ellipse C to the focus is 3. (I) find the ellipse
It is known that the center of the ellipse C is at the coordinate origin, the focus is on the X axis, and the eccentricity is
The maximum distance from the point on the ellipse C to the focus is 3
(I) to find the standard equation of ellipse C;
(II) if the line L passing through the point P (0, m) intersects the ellipse C at two different points a and B, and
AP=3
Pb, find the value range of real number M
It is known that the center of the ellipse C is at the origin of the coordinate, the focus is on the x-axis, the eccentricity is 1 / 2, and the distance from the left vertex of the ellipse to the right focus is 3
(2) If the line L passing through the point P (0, m) intersects the ellipse C at two different points a, and the vector AP = vector 3PB, find the value range of the real number M.

(1) Let the standard equation of ellipse be: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1
　　∵e=1/2 ∴c=√(a^2-b^2)=a/2
∵ the distance from the left vertex to the right focus is 3
　　∴a+c=3 a+a/2=3 a=2 c=1 b=√3
The standard equation of ellipse is: x ^ 2 / 4 + y ^ 2 / 3 = 1
(2) Let the equation of a straight line passing through a point m + m be
And x ^ 2 / 4 + y ^ 2 / 3 = 1
　　(-2[2km-√(9-3m^2+12k^2)]/(3+4k^2),[2k√(9-3m^2+12k^2)+3m]/(3+4k^2))
　　(-2(2km+√(9-3m^2+12k^2)]/(3+4k^2),[-2k√(9-3m^2+12k^2)+3m]/(3+4k^2))
　　AP=(-1/(3+4k^2)*[-4km-2√(9-3m^2+12k^2)],-k/(3+4k^2)*[-4km-2√(9-3m^2+12k^2)])
　　PB=( 1/(3+4k^2)*[-4km+2√(9-3m^2+12k^2)],k/(3+4k^2)*[-4km+2√(9-3m^2+12k^2)])
∵ vector AP = vector 3PB
　　∴-1/(3+4k^2)*[-4km-2√(9-3m^2+12k^2)]=3/(3+4k^2)*[-4km+2√(9-3m^2+12k^2)]
-k/(3+4k^2)*[-4km-2√(9-3m^2+12k^2)]=3k/(3+4k^2)*[-4km+2√(9-3m^2+12k^2)]
　　∴4km+2√(9-3m^2+12k^2)=-12km+6√(9-3m^2+12k^2)
4km=√(9-3m^2+12k^2)
16k^2m^2=9-3m^2+12k^2
(16k^2+3)m^2=9+12k^2
m^2=(9+12k^2)/(3+16k^2)