Finding the positive integer solution of the equation x square - y square = 24

Finding the positive integer solution of the equation x square - y square = 24

Junior one should have learned integral multiplication and binary linear equations related knowledge
Note that X & # 178; - Y & # 178; = (x + y) (X-Y) = 24 = 24 × 1 = 12 × 2 = 8 × 3 = 6 × 4
Because it is a positive integer solution, the possible case is
x+y=24,x-y=1;
x+y=12,x-y=2;
x+y=8,x-y=3;
x+y=6,x-y=4;
Respectively solve the above four equations can be suitable for the problem of positive integer solution is
x=7,y=5
Or x = 5, y = 1
There are only two cases

Is there an integer solution to the equation x squared minus y squared = 1990

1990=2*5*199
And the square of x minus the square of y = (x + y) (X-Y)
(x + y) and (X-Y) are the same parity
It can only be 10 * 199 or 398 * 5 or 2 * 955
It is impossible to be parity, so there is no integer solution

If the chord length of the straight line passing through the point (- 1, - 2) cut by the circle x + y-2x-2y + 1 = 0 is √ 2, then the slope of L

Let the slope of the line l be K, then the equation of the line L passing through the point (- 1, - 2) is y + 2 = K (x + 1), that is y = KX + K-2, by the square of the circle x + the square of Y - 2x-2y + 1 = 0, that is, (x-1) &# 178; + (Y-1) &# 178; = 1, that is, the center of the circle is (1,1) and the radius is 1

What is the minimum value of y = lgx / 3lgx / 12 when x is
Lg3 x LG12 x

Because: find the minimum value of LG (x / 3) LG (x / 12), so this formula should be meaningful
Then x > 0
Because LG (x / 3) LG (x / 12) = (lgx-lg3) (lgx-lg12) = (lgx) ^ 2 - (Lg3 + LG12) lgx + lg3lg12
=(lgx)^2-lg36lgx+lg3lg12=(lgx)^2-2lg6lgx+(lg6)^2-(lg6)^2+lg3lg12
=(lgx-lg6) ^ 2 - (LG6) ^ 2 + lg3lg12 (where (lgx) ^ 2 is the square of lgx)
To minimize, then (lgx-lg6) ^ 2 = 0, so x = 6
In this case, the minimum value is lg3lg12 - (LG6) ^ 2 = (LG2) ^ 2

A classroom is 8 meters long, 6 meters wide and 4 meters high. The four walls and ceiling of the classroom should be painted. The area of doors and windows outside is 24 square meters,
4 kg / m2, 2.5 yuan / kg, 5 yuan / m2
(1) How many kilos of paint does this classroom share?
(2) How much is the cost and labor for painting this classroom?
Line will. A total of two questions, some trouble you play, three grams of oil

The area is 8 × 6 + (8 + 6) × 2 × 4-24 = 136 square meters
So 136 × 0.4 = 54.4kg
Labor cost 136 × 5 = 680 yuan
Cost 54.4 × 2.5 = 136 yuan

Given the ellipse x ^ 2 / 16 + y ^ 2 / 4 = 1, find the trajectory equation of the midpoint of the chord whose slope is 3
process

Let the coordinates of string AB be a (x1, Y1), B (X2, Y2)
Midpoint m (x, y)
The results are: X1 + x2 = 2x, Y1 + y2 = 2Y
(Y1-Y2)/(X1-X2)=3.
A. The B coordinate is substituted into:
X1^2/16+Y1^2/4=1
X2^2/16+Y2^2/4=1
By subtracting the two formulas, we get the following result:
(X1+X2)(X1-X2)/16+(Y1+Y2)(Y1-Y2)/4=0
2X/(X1-X2)/16=-2Y(Y1-Y2)/4
X=4Y*(Y1-Y2)/(X1-X2)
So, x = 4Y * 3 = 12Y

Two triangles of equal area can form a parallelogram______ (judge right or wrong)

For example: two triangles with the length of 4 and the height of 3 and the length of 2 and the height of 6 have the same area, but they cannot be combined into parallelogram. Two triangles with the same area can be combined into parallelogram, which is wrong

Given the fixed point F (1,0), the moving point P (different from the origin) moves on the y-axis, connects PF, crosses point P as PM, intersects the x-axis at point m, and extends MP to point n
Given the fixed point F (1,0), the moving point P (different from the origin) moves on the y-axis, connects PF, crosses point P as PM, intersects the x-axis at point m, and extends MP to point n, and the vector PM * vector pf = 0, vector | PN | = vector | PM|
1. Find the equation of LOCUS C of moving point n
How to see: n is the symmetric point of m with respect to P

Because vector PM * vector pf = 0,
∴MP⊥PF
That is Mn ⊥ PF
And vector | PN | = vector | PM|
∴PN=PM
The symmetry of M, n with respect to p
N is the symmetric point of m with respect to P

First simplify, then evaluate: 12m-2 (m − 13n2) - (32m − 13n2), where M = 13, n = - 1

When m = 13, n = - 1, the original formula = - 3 × 13 + (- 1) 2 = 0

As shown in the figure, PA is the tangent line of ⊙ o, and a is the tangent point. The straight line Po and ⊙ o intersect at two points B and C, with ∠ P = 30 ° connecting Ao, AB and ac. verification: △ ACB ≌ △ apo

It is proved that ∵ PA is the tangent of ⊙ o, and ∵ Pao = 90 degrees, and ∵ P = 30 degrees, and ∵ AOP = 60 degrees, and ∵ OA = OC, and ∵ C = OAC, and ∵ AOP = 12, AOP = 30 degrees, and ∵ C = P, AC = AP. BC is the diameter of ⊙ o, and ∵ cab = Pao = 90 degrees, and ≌ ACB ≌ apo (ASA)