The left and right focus of hyperbola F1F2, x ^ 2-y ^ 2 / 9 = 1, point P on hyperbola, vector Pf1 * PF2 = 0, find the absolute value of vector Pf1 + PF2

The left and right focus of hyperbola F1F2, x ^ 2-y ^ 2 / 9 = 1, point P on hyperbola, vector Pf1 * PF2 = 0, find the absolute value of vector Pf1 + PF2

X²-Y²/3²=1==>C=√[1+3²]=√10.
According to the parallelogram rule of vector, 2 vector Po = vector Pf1 + vector PF2
In RT Δ pf1f2: OP = of1 = of2 = √ 10
| vector Pf1 + vector PF2 | = 2 √ 10

It is known that 64 / x square minus 36 / y square of hyperbola is equal to 1, the focus is F1F2, and Pf1 is perpendicular to F2 to find the area of triangle f1pf2

If ofl & sup2; = 6 & sup2; + 8 & sup2; = 10 & sup2;. | F1F2 | = 20. X = 10, y = 36 / 8. (P ∈ hyperbola, missing)
S ⊿ f1pf2 = 20 × (36 / 8) / 2 = 360 / 8 = 45 (area unit)

It is known that P is the square of hyperbola x divided by 12 minus the square of Y divided by 4, which equals to a point on one. F1F2 is the left and right focus of hyperbola, and the angle f1pf2 is equal to 120 degrees. Calculate the area of angle f1pf2

First, we use the cosine theorem to have the relationship between Pf1 and PF2, which is the relationship 1, and the absolute value of the difference between Pf1 and PF2 is 2a, which is the relationship 2. We can get the length of Pf1 and PF2 by simultaneous one or two, or you can directly multiply them, and then the area is equal to half times the product, and then multiplied by the sine value of 120 degrees!

hyperbola
If the distance between the perpendicular foot of the focus of the hyperbola on its asymptote and the origin is equal to the length of the imaginary half axis, the eccentricity of the hyperbola is calculated?

If the distance between the perpendicular foot of the focus on the asymptote and the origin of the hyperbola is equal to the length of the imaginary half axis, find the eccentricity of the hyperbola? (1). Let the focus be on the X axis, then the equation of an asymptote is y = (B / a) x, that is, ay BX = 0, and the distance from the right focus f (C, 0) to the asymptote is h = ∣ - BC ∣ / √ (A & # 178; + B & # 178;) = BC / √ C

Who knows the simplest way to get the asymptote of hyperbola?
Find it in the simplest way!

Directly change X & sup2; / A & sup2; - Y & sup2; / B & sup2; = 1 to
X & sup2 / / A & sup2; = y & sup2 / / B & sup2;, this is the equation of those two asymptotes
Y & sup2 / A & sup2; - X & sup2 / B & sup2; = 1
y²/a²=x²/b²

What is the definition of asymptote in hyperbola

When a point m on the curve is infinitely far away from the origin along the curve, if the distance from m to a straight line infinitely approaches zero, then the straight line is called the asymptote of the curve. If the limit exists, and the limit also exists, then the curve has the asymptote y = ax + 1

What effect does the asymptote of hyperbola have on the opening of hyperbola?

For example, the hyperbola x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1, the larger the B / A, the larger the opening of the hyperbola

On the asymptote of hyperbola
Why can we say that the standard formula of hyperbola is changed from 1 to 0 to seek asymptote? What does this mean?

x²/a²-y²/b²=1
The asymptote here means that at infinity there is an intersection between the line y = KX and the hyperbola
So simultaneous equations
We obtain B & sup2; X & sup2; - A & sup2; K & sup2; x-a & sup2; B & sup2; = 0
That is, k = ± B / A

How to judge whether the equation is hyperbola or ellipse?
Their equations are very similar

It's the sign of the standard equation
Hyperbola: X & sup2 / / a-y & sup2 / / B & sup2; = 1
Ellipse: X & sup2 / / A & sup2; + Y & sup2 / / B & sup2; = 1
It is obvious that there is no "-" sign in elliptic equation, but hyperbola has

In the standard equation of ellipse, parabola and hyperbola, which part does it represent after changing the equal sign to greater than or equal to, and less than or equal to
rt

In the standard equation of ellipse, changing the equal sign to greater than sign indicates the point outside the ellipse, changing the equal sign to less than sign indicates the point inside the ellipse; in the standard equation of parabola, changing the equal sign to greater than sign indicates the point outside the parabola, changing the equal sign to less than sign indicates the point inside the parabola; hyperbolic