Given that P is any point of hyperbola x ^ 2 / 2-y ^ 2 = 1, find the minimum distance between point a (m, 0) (M > 0) and point P?

Given that P is any point of hyperbola x ^ 2 / 2-y ^ 2 = 1, find the minimum distance between point a (m, 0) (M > 0) and point P?

Let the coordinates of point p be (x, y), then | AP | = √ [(x-m) ^ 2 + y ^ 2] ①
From x ^ 2 / 2-y ^ 2 = 1 to y ^ 2 = x ^ 2 / 2-1 ②
Take the second generation and the first generation and get | AP | = √ [2 / 3 * x ^ 2-2mx + m ^ 2-1], x > = √ 2
Let f (x) = 2 / 3 * x ^ 2-2mx + m ^ 2-1
The symmetry axis of quadratic function f (x) is x = 2m / 3
When 0

Given that x 2 of 16 + y 2 of 4 of an ellipse is 1, (1) if a chord ab of the ellipse is bisected by point m (1,1), find the linear equation of ab

Let a point (x1, Y1) and B (X2, Y2) be substituted into the equation, where 1 / 16 (x1) + 4 / 4 (Y1) = 1 / 1 formula, 16 / 16 (x2) + 4 / 4 (Y2) = 1 / 2 formula, 1 formula minus 2 formula,
(x1 + x2) (x1-x2) / 16 + (Y1 + Y2) (y1-y2) / 4 = 0, and (x1 + x2) = 2 (Y1 + Y2) = 2
So (x1-x2) + 4 (Y1 + Y2) = 0, then (Y1 + Y2) / (x1-x2) = - 1 / 4, that is, k = - 1 / 4, and the linear equation x + 4y-5 = 0

It is known that the ellipse C: x2 / A2 + Y2 / B2 = 1 (a > 0, b > 0) passes through the point (1,2 / 3), and the eccentricity is 1 / 2

When a > b, the x-axis of the focus is the same
Centrifugation e = C / a = 1 / 2 a = 2C
A ^ 2 + B ^ 2 = C ^ 2, so B ^ 2 = 3C ^ 2
X2 / A2 + Y2 / B2 = 1, that is, X2 / 4C2 + Y2 / 3c2 = 1
Substituting (1,2 / 3), C = √ 129 / 18
The equation is 27x ^ 2 / 43 + 36Y ^ 2 / 43 = 1
a

It is known that the equation of ellipse is x2 / A2 + Y2 / B2 = 1 (a > b > 0)

Given that a focus of the ellipse x2 / A2 + Y2 / B2 = 1 (a > b > 0) is (√ 6,0), and the eccentricity is √ 3 / 2, then the equation of the ellipse is

The focus of solution is (√ 6,0)
That is, C = √ 6
And E = C / a = √ 6 / a = √ 3 / 2
The solution is a = 2 √ 2
So B ^ 2 = a ^ 2-C ^ 2 = 8-6 = 2
So the elliptic equation is
x2/8+y2/2=1