The tangent problem of the conic curve, the conclusion of the tangent equation of the conic curve

When there is only one intersection, that intersection is a tangent. Think about the circle... This is not the case for hyperbolas, because when the line is parallel to the asymptotic line, there may also be only one intersection point. The difference is that an ellipse is a closed quadratic curve, while a hyperbola is not.

When I asked the teacher, he said that if the linear equation of a point is a quadratic equation obtained after concatenation, the k obtained when the quadratic coefficient is 0" This sentence is not too clear, it is recommended that you take a good look at the definition of tangent by yourself. Actually, what I'm talking about is also understanding... And not strict proof.

This question is about derivation.

If the point p(m,n) is in the ellipse.

x^2/a^2+y^2/b^2=1

on, then. The tangent equation for crossing the point p is.

m/a^2)x+(n/b^2)y=1

Hyperbola and parabola. Similar.

For example, x 2 a 2+y 2 b 2=1 over the fixed point (2,3) then replace one x in the equation with 2 and one y with 3

That's it!

High school students set up lines over points and solve systems of equations, and the discriminant formula is 0

College students seek guidance and k

Encyclopedia definition: "P and Q are two adjacent points on the curve C, P is the fixed point, when the point Q is infinitely close to the point P along the curve C, the limit position of the secant Pq is called the tangent of the curve C at the point P, and the point P is called the tangent point."

In plane geometry, a straight line that has only one common intersection with a circle is called a tangent of a circle, and this definition does not apply to curves in generalpt is the tangent of the curve c at the point p, but it has another intersection with the curve c;Conversely, the straight line l, although it has only one intersection point with the curve C, is not a tangent of the curve C".

In the sense of a conic curve, there is only one point of intersection, which is the tangent. (For hyperbola, "only one intersection" means that one has an intersection and the other ignores.) ）

But there is actually a more convenient way to find tangent equations. This is the calculus, the differentiation of calculus.

You can teach yourself.

I just wanted to help you calculate the formula for each conic curve. I found out that something went wrong at the beginning. So it led to a miscalculation later. I'll tell you next time it's done. It's time to go to bed now

- The following modifications ---

Correct my fallacy. As stated on the 2nd floor, when the line is parallel to the hyperbola, there is only one intersection point, but not a tangent point. Moreover, when the slope of the straight line does not exist, there can also be an intersection point, which cannot be calculated by the method of simultaneous equations.

There is also a parabola in which the straight line parallel to the axis of the parabola also has only one intersection point with the parabola, but it is not a tangent.

- Start calculating for you. -

For an ellipse, let its equation be (x-m) 2 a 2+(y-n) 2 b 2=1 (1).

In general form: b 2(x-m) 2+a 2(y-n) 2=a 2b 2

Derivative of x on both sides: b 2*2(x-m)+a 2(2(y-n)*y')=0

Simplification will be y'Put it on the side of the equation alone to get equation (2), and solve y in (1) and bring in (2) to get equation y'=f(x)

In this way, when a tangent is required at a certain point, X is brought into Y'=f(x) y'This is the slope of the tangent at this point.

The tangent equation can be determined by using the point oblique formula.

over.Calculus is very useful, and it is recommended that you learn some. You look like you're just a high school student.

But I'm also = - learn a little, physics competitions are inseparable.

The tangent equation of the conic curve concludes that x 2 a 2 + y 2 b 2 = 1.

Conic curves include ellipses, hyperbolas, parabolas.

1. Ellipse: The trajectory of a moving point whose sum of the distances to two fixed points is equal to the fixed length (the fixed length is greater than the distance between the two fixed points) is called an ellipse. Namely:.

2. Hyperbola: The absolute value of the difference between the distance to the two fixed points is a fixed value (the fixed value is less than the distance between the two fixed points) and the moving point trajectory is called hyperbola. Namely:.

3. Parabola: A moving point trajectory with an equal distance to a fixed point and a fixed straight line is called a parabola.

4. Unified definition of conic curve: the ratio of the distance to the fixed point to the distance to the fixed line e is constant, and the trajectory of the point is called the conic curve. When 0<1 is an ellipse: when e=1 is a parabola; When e>1 is hyperbola.

Definition of three-dimensional geometry: The geometry enclosed by the curved surface formed by the straight line where the right-angled side of the right-angled triangle is the axis of rotation, and then the other two sides rotate 360 degrees is called a cone. The axis of rotation is called the axis of the cone.

Second, the surface that rotates perpendicular to the edge of the axis is called the base surface of the cone.

1.Let the tangent equation be y-1=k(x-1), substitute the curve equation, and use the discriminant formula of the quadratic equation = 0 to determine k

2.To derive the curve equation, 1) the tangent equation can be written from the geometric properties of the derivative if the point is known to be on the curve;

2) If the known point is not on the curve, assuming that the tangent point is (x0, y0), write the tangent equation, and then substitute the coordinates of the known point. For example.

The point a(1,1) is not on 2x +y =1 and the derivative of is 4x+2yy'=0,y'=-2x y, let the tangent equation be y-y0=(-2x0 y0)(x-x0), where (x0,y0) satisfies 2x0 +y0 =1, becomes 2x0x+y0y=1, it passes the point a, 2x0+y0=1,y0=1-2x0, and the solution of x0,y0 is reduced to (1).

The specific calculations are left to you to practice.

Let the f coordinate (c,0) and the slope of the asymptotic line is k=b a or -b a.

Then the slope of Fa is k'=-a/b.

The fa equation is y=-a b(x-c).

Simultaneous y=-b a x, the solution is x=c+b 2 a 2, y=-b a(c+b 2 a 2)=-bc a+b 3 a 3

From the meaning of the title, af=ab, so get: of=ob

i.e. c 2 = (c + b 2 a 2) 2 + (-bc a + b 3 a 3) 2

Simplification: 0=b 2 a 4+2c a 2+c 2 a 2-2b 2 a 4+b 4 a 6

0=-a^2b^2+2a^4c+a^4c^2+b^4

b^2=a^2-c^2

The solution can be solved.

The two roots of the equation 2x 2-5x+2=0 are: 1 2,2 The conic curve includes an ellipse, a hyperbola, a parabola (eccentricity of 1, rounded) and eccentricity: e=c a

mx 2+4y 2=4m into a standard shape: x 2 4+y 2 m=1 if e = 1 2, then it is an ellipse, m>0

If m>4, then a 2 = m, b 2 = 4, c 2 = m-4, then: (m-4) m = 1 4, solution: m1

If 0 if e=2, it is hyperbola, m<0

and a = 4, b 2 = -m, c 2 = 4-m

4-m) 4=2, and the solution is m3=-4

So it's 3 of a kind.

Let ·y=kx+b, substitute the coordinates of the point, the simultaneous curve equation, and eliminate the element, and obtain the binary linear equation. Make diao ta (discriminant) = 0

You can find the value of x,y. When we add back ·y=kx+b, we can find the tangent equation. Find the tangent equation for all points on the conic curve, and the tangent equation for a point outside the conic curve. (Try it yourself).