It is known that there are two intersections between the straight line y = 2x + m and the ellipse x ^ 2 / 9 + x ^ 2 / 4 = 1

It is known that there are two intersections between the straight line y = 2x + m and the ellipse x ^ 2 / 9 + x ^ 2 / 4 = 1

y=2x+m
Substituting 4x & sup2; + 9y & sup2; = 36
40x²+36mx+9m²-36=0
If there are two intersections, the equation has two solutions
The discriminant is greater than 0
1296m²-1440m²+5760>0
m²

It is known that the straight line y = 2x + m and the ellipse x square ratio 9 + y square ratio 4 = 1, the intersecting chord length is 3. Find the real number M

The straight line y = 2x + m and ellipse x ^ 2 / 9 + y ^ 2 / 4 = 1 are combined
45X^2+36XM+9M^2-36=0.
The difference between the two is (- 4m / 5) ^ 2-4 * [(m ^ 2-4) / 5] = 3 / root 5 (chord length formula. Weida theorem)
The solution is enough

Given that the nonnegative real numbers x, y, Z satisfy x − 12 = 2 − Y3 = Z − 34, w = 3x + 4Y + 5Z. Find the maximum and minimum of W

Let x − 12 = 2 − Y3 = Z − 34 = k, then x = 2K + 1, y = - 3K + 2, z = 4K + 3, ∵ x, y, Z are nonnegative real numbers, ∵ 2K + 1 ≥ 0 − 3K + 2 ≥ 04k + 3 ≥ 0, the solution is - 12 ≤ K ≤ 23, then w = 3x + 4Y + 5Z = 3 (2k + 1) - 4 (3K-2) + 5 (4K + 3) = 14K + 26, ∵ - 12 × 14 + 26 ≤ 14K + 26 ≤ 23 × 14 + 26, that is 19 ≤ w ≤ 3513