Find the chord length of the line y = 2x-1 cut by the ellipse x ^ 2 / 4 + y ^ 2 / 9 = 1

Here are two ways
Method 1: connect the analytical expression of straight line and ellipse to get the intersection point, and then use the distance formula between two points to get the chord length
The analytic method of ellipse is different from that of quadratic equation
On X: 25X ^ 2-16x-32 = 0
On Y: 25y ^ 2 + 18y-135 = 0
According to the formula of distance between two points
Let the endpoint of the chord be a (x1, Y1) B (X2, Y2)
Then the square of chord length AB = (x1-x2) ^ 2 + (y1-y2) ^ 2
According to Weida's theorem, we get
(x1+x2)^2-4x1x2+(y1+y2)^2-4y1y2
Get by substitution
(16 / 25) ^ 2 + 128 / 25 + (- 18 / 25) ^ 2 + 540 / 25 = 24 times root 30 / 25
Among them, I recommend method 2. Many big problems have to use this technique

What is the absolute value of X + y = 4 xy = - 2 X-Y?

2√6

It is known that the coordinates of the intersection points of the straight lines x-2y-2k = 0 and 2x-3y-k = 0 are the solutions of the equation x ^ 2 + y ^ 2 = 25, and the value of K is obtained

Through the simultaneous equations, the intersection coordinates of two straight lines are obtained
( -4k ,-3k)
Substituting into the equation of the circle, we get k = + - 1

How to use the pinch rule?

For example, it is required to find an equation x (generally, it is to find the limit). First, find an equation a whose limit is larger than it and an equation B whose limit is smaller than it, and then prove that limit a is equal to limit B

The teacher assigned a problem to calculate (x + y) (x-2y) - my (nx-y) (m, n are constants)
Calculate (x + y) (x-2y) - my (nx-y) (m, n are constants)
When substituting the values of X and Y into the calculation, careless Xiaoming and Xiaoliang misjudged the value of Y, but the results were all equal to 25. Careful Xiaohong substituted the correct values of X and Y into the calculation, and the result was just 25. In order to find out, she randomly changed the value of y into 2011, and the result was still 25
(1) According to the above situation, try to explore the mystery
(2) Can you determine the values of M, N and X?

∫ (x + y) (x-2y) - my (nx-y) = x & # 178; - (1 + Mn) XY + (m-2) y & # 178;, and the original formula has nothing to do with the value of Y, we can judge that m-2 = 0, (1 + Mn) = 0. In this case, the value of the original formula = x & # 178; has nothing to do with the Y axis. (2) because the value of the original formula has nothing to do with the Y axis, m-2 = 0, M = 2, (1 + Mn) = 0, n = - 1 / 2

Questions about the criterion of pinch higher mathematics
I remember that the pinch rule is that the function a is less than or equal to the function B is less than or equal to the function C, and then the limit of a = C is equal, then the limit of B is also equal. Can the "less than or equal" be changed into "less than" also holds? I see that it is written in some reference books

Theorem has equal sign
In the actual use of time, depending on the situation, there can be no equal sign

Xiaoming, Xiaoliang and Xiaomin are calculating the values of (x + y) (x-2y) - my (nx-y) (m, n are all constants)
Careless Xiao Ming and Xiao Liang misjudge the value of Y, but the result is equal to 25. Careful Xiao Hong substitutes the correct values of X and Y into the calculation, and the result is just dead 25. In order to find out, she randomly changes the value of Y into 2006, and the result is still 25

(x + y) (x-2y) - my (nx-y) = x & # 178; - (1 + Mn) XY + (m-2) y & # 178; - (1 + Mn) XY + (m-2) y & # 178; - (1 + Mn) x + m-2 = 25 (1) x & # 178; - (1 + Mn) x + m-2 = 25 (1) x & # 178; = 25 (2) the solution (2) is x = ± 5, and x = 5 and x = - 5 are substituted into (1) the system of equations m-5 respectively

Advanced mathematics, on the use of the pinch criterion
If I can only prove a = g (x)

Instead of F (x) = a, the limit of F (x) is equal to a
Generally, the pinch theorem can't get equal sign

Calculate the value of (x + y) (x-3y) - my (nx-y) (m.n are all constants). When substituting the value of X and Y into the calculation, careless Xiaoming and Xiaoliang misjudged the value of Y, but the calculation results were all equal to 25. Careful Xiaomin substituted the correct value of X and Y into the calculation, and the result was exactly 25. In order to find out, she arbitrarily took 2012 as the value of Y. you say no wonder, the result was still 25
Try to explore the mystery;
Determine the values of M, N and X

The original formula = (x + y) (x-3y) - my (nx-y) = x ^ 2-3xy + xy-3y ^ 2-mnxy + my ^ 2 = x ^ 2 - (2 + Mn) XY + (M-3) y ^ 2
When M-3 = 0, Mn + 2 = 0, i.e. M = 3, n = - 2 / 3, no matter what the value of Y is, the result is only related to x, not to y
That is, x ^ 2 = 25, x = (+ / -) 5

Using the pinch theorem to find the limit of this problem,
Find the limit of (n →∞) Lim [√ 1 ^ 2 + 2 ^ 2 + 3 ^ 2 +. + n ^ 2] / n

The original formula = (n →∞) Lim [√ n (n + 1) (2n + 1) / 6] / N > = (n →∞) Lim [√ n ^ 3 / 6] / N = (n →∞) Lim [√ n / 6] = + ∞
also
simple form

Find the chord length of the line y = 2x-1 cut by the ellipse x ^ 2 / 4 + y ^ 2 / 9 = 1