The elliptic equation with focus on x-axis, focal length of 4 and eccentricity of 1 / 2 is

If the maximum area of a point P and two triangles with vertex focus on the ellipse is 1, then the minimum length of the major axis of the ellipse?

First, the maximum area is obtained at the end of the minor axis, i.e. s_ max=bc=1.
So the problem is to minimize a = (b ^ 2 + C ^ 2) ^ 0.5 when BC = 1, b > 0, C > 0
From the basic inequality a > = (2BC) ^ 0.5 = 2 ^ 0.5, when B = C = 1, we take the equal sign
So the minimum value of the length of the major axis is the root 2

The area of the triangle formed by one end point and two focal points of the minor axis of the ellipse is 12, and the distance between the two directrix is 25 / 2

There are indeed two sets of solutions, and the second set of solutions is really
It is estimated that the author made a mistake when he put together the figures
In fact, the binary quadratic equations can easily be transformed into one variable quartic equations (all binary quadratic equations are eliminated no more than four times), and a solution can be seen at a glance. Then, the cubic equation has no rational root. Although it can be judged that there is a real root and it is greater than 0, it can only replace Cardano formula

If the maximum area of a triangle with one point and two focus points on the ellipse is 1, then the minimum length of the major axis is ()?
2√2
Why?

The maximum area of a triangle is 1
Bottom = 2C
High = 2BC = 2
a>√2
Long axis length = 2A > = 2 √ 2

If the area of the triangle with one point on the ellipse and two focal points of the ellipse as the vertex is 8, what is the minimum length of the major axis of the ellipse?

The area of triangle is ch = 8
0

Hyperbola why the greater the eccentricity, the greater the opening!

The asymptote slope of hyperbola with focus on X axis is: B / a = under the root sign (square of E-1)
The larger the E is, the larger the slope of the asymptote is, the larger the opening angle of the two asymptotes is, and the larger the opening of the hyperbola is
The asymptote slope of hyperbola with focus on Y-axis is: A / b = 1 / root (square of E-1)
The larger e is, the smaller the slope of the asymptote is, the smaller the opening angle of the two asymptotes is, and the larger the complementary angle of the opening angle (the angle with hyperbola) is
Therefore, regardless of whether the focus is on the X or Y axis, the greater the eccentricity, the larger the opening

Why does the eccentricity affect the opening size of hyperbola
Using the hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1, the larger the opening is, instead of comparing the size of y when x is the same, the larger the opening is. What does it have to do with hyperbolic eccentricity or B / a. I don't think it's accurate to judge by asymptote

E = C / a = radical (a ^ 2 + B ^ 2) / A ^ 2 = radical 1 + (b ^ 2 / A ^ 2)
Because B / A is the asymptote slope, when e is larger, B / A is larger, and the asymptote slope is larger, so the opening is larger

The length and focal distance of the real axis and imaginary axis of a hyperbola form an arithmetic sequence to calculate the eccentricity of the hyperbola

a^2+b^2=c^2
a+c=2b
5a^2=3c^2-2ac
5=3e^2-2e
E1 = 5 / 3 E2 = - 1 (incompatible)

How to make the second definition of conic with Geometer's Sketchpad
If the ratio of the distance to a fixed point to the distance to a fixed line is constant (eccentricity), the conic will change with the change of eccentricity

Step 1 draw a line K2 draw a point A3 select point a and line k, construct / parallel line J4 select line k, construct / point on line B5 select midpoint B and line k, construct / vertical line L6 select vertical line L, construct / point on vertical line O7 select vertical line L, construct / point on vertical line P8 select

Focus chord problem of conic
I'm a liberal arts student, so we didn't learn something about science
What is the nature of the focus chord in conic? What is their formula?
A focus chord of hyperbola is called path. What is the path? Is it the focus chord perpendicular to the X axis? Is there path in ellipse and parabola, or is it called path?
Please, thank you

The path of a hyperbola is a chord passing through the focus and perpendicular to the real axis

The elliptic equation with focus on x-axis, focal length of 4 and eccentricity of 1 / 2 is