Given that x ^ 2 + y ^ 2 = 1 and the line y = 2x + m intersect a and B, and the angles between OA, OB and the positive direction of X axis are a and B respectively, it is proved that sin (a + b) is a fixed value

Given that x ^ 2 + y ^ 2 = 1 and the line y = 2x + m intersect a and B, and the angles between OA, OB and the positive direction of X axis are a and B respectively, it is proved that sin (a + b) is a fixed value

Let x = cosx, y = SiNx, SiNx = 2cosx + M
Sin (a + b) = sinacosb + sinbcosa, SiNx is replaced by cosx
sin(a+b)=4cosacosb+M(cosa+cosb)
Substituting SiNx = 2cosx + m into SiNx ^ 2 + cosx ^ 2 = 1, 5cosx ^ 2 + 4mcosx + m ^ 2-1 = 0
According to Weida's theorem, sin (a + b) = 4 (m ^ 2-1) / 5-4m ^ 2 / 5 = - 4 / 5

It is known that the intersection of a straight line y = 2x + m and a circle x2 + y2 = 1 is at two different points a and B, and the angles with ox as the starting edge, OA and ob as the ending edges are α and β respectively, then the value of sin (α + β) is ()
A. 35B. −45C. −35D. 45

As shown in the figure, if O is used as OC ⊥ AB at point C, OC bisects ⊥ AOB, because the angles of ox as the starting edge, OA and ob as the ending edge are α and β respectively, so ⊥ AOD = α, ⊥ BOD = β, so ⊥ cod = α + β 2 is OC ⊥ AB, and the slope of AB is K1 = 2, so the slope of OC is K2 = - 12, so tan α + β 2 = - 12

It is known that the intersection of a straight line y = 2x + m and a circle x2 + y2 = 1 is at two different points a and B, and the angles with ox as the starting edge, OA and ob as the ending edges are α and β respectively, then the value of sin (α + β) is ()
A. 35B. −45C. −35D. 45

As shown in the figure, if O is used as OC ⊥ AB at point C, OC bisects ⊥ AOB, because the angles of ox as the starting edge, OA and ob as the ending edge are α and β respectively, so ⊥ AOD = α, ⊥ BOD = β, so ⊥ cod = α + β 2 is OC ⊥ AB, and the slope of AB is K1 = 2, so the slope of OC is K2 = - 12, so tan α + β 2 = - 12. According to the universal formula, sin (α + β) = 2 × (− 12) 1 + (− 12) 2 = - 45

The line L passing through point m (1,2) and circle C: (X-2) 2 + y2 = 9 intersect at two points a and B, C is the center of the circle, when ∠ ACB is the minimum, the equation of line L is______ ．

∵ the equation of circle C is: (X-2) 2 + y2 = 9, ∵ the coordinate of center C is (2,0), radius r = 3. ∵ point m (1,2) is a point inside circle C, the line L passes through point m (1,2) and intersects with circle C at two points a and B. according to the properties of circle, when cm is perpendicular to L, the chord length AB is the shortest, and correspondingly ∠ ACB is the smallest

The equation of the line L passing through point P (2,1) and the circle C: (x-1) ² + Y & #178; = 4 intersect at two points a and B, when the angle ACB is minimum, the equation of the line L
The line L passing through the point P (2,1) intersects with the circle C: (x-1) ² + Y & #178; = 4 at two points a and B. when the angle ACB is the smallest, the equation of the line L is (). The chord length | ab | = ()

Point (2,1) is in the circle, Center (1,0)
The slope of the line passing through the center of the circle and point P (2,1) is: (1-0) / (2-1) = 1
When line L is perpendicular to line n

The line L passing through point m (12,1) intersects with circle C: (x-1) 2 + y2 = 4 at two points a and B. C is the center of the circle. When ∠ ACB is the minimum, the equation of line L is______ ．

Verify that the known point m (12,1) is in the circle. When ∠ ACB is the smallest, the straight line L is perpendicular to cm. According to the equation of the circle, the center of the circle C (1,0) ∵ KCM = 1 − 012 − 1 = - 2, ∵ KL = 12 ∵ l: Y-1 = 12 (X-12), we get 2x-4y + 3 = 0, so we should fill in 2x-4y + 3 = 0

In the circle determined by the equation x2 + Y2 + X + (m-1) y + 12m2 = 0, the maximum area is ()
A. 32 π B. 34 π C. 3 π D. does not exist

The formula of the equation is (x + 12) 2 + (y + m − 12) 2 = − (M + 1) 2 + 34. R2max = 34, where M = - 1. The maximum area is 34 π

The right branch of the straight line y = KX + 1 and the hyperbola C: 2x ^ 2-y ^ 2 = 1 intersect at two different points a and B. if the right focus F of the hyperbola C is on a circle with diameter AB, find K

a= √2/2,b=1,c=√6/2,
The right focus F of hyperbola C is on a circle with diameter ab,
Then AF ⊥ BF,
Let a (x1, Y1), B (X2, Y2),
F(c,0),
Vector AF = (x1-c, - Y1),
Vector BF = (x2-c, - Y2),
⊥ vector AF ⊥ BF
The vector AF · BF = x1x2-c (x1 + x2) + C ^ 2 + y1y2 = 0,
y1=kx1+1,
y2=kx2+1,
y1y2=k^2x1x2+k(x1+x2)+1,
x1x2-c(x1+x2)+c^2+k^2x1x2+k(x1+x2)+1=0,
x1x2(1+k^2)+(x1+x2)(k-c)+1+c^2=0,
The linear equation is replaced by the hyperbolic equation,
2x^2-(kx+1)^2=1,
(2-k^2)x^2-2kx-2=0,
According to Veda's theorem,
x1+x2=2k/(2-k^2),
x1x2=-2/(2-k^2),
(1+k^2)*[-2/(2-k^2)]+2k(k-c)/(2-k^2)+1-c^2=0,
-2-2k^2+2k^2-√6k+(1-3/2)(2-k*2)=0,
k^2-2√6k-6=0,
∴k=(√6±2√15)/2.

If it is known that the circle C is tangent to the line X-Y = 0 and X-Y-4 = 0, and the center of the circle is on the line x + y = 0, then the equation of circle C is______ ．

∵ the center of a circle is on the straight line x + y = 0, let the coordinates of the center of a circle be (a, - a) ∵ the center of a circle C is tangent to the straight line X-Y = 0. The distance between the center of a circle (a, - a) and the two straight lines X-Y = 0 is: | 2A | 2 = R

It is known that the circle C passes through points a (- 2,0), B (0,2), and the center of the circle C is on the straight line y = x, and the straight line L: y = KX + 1 intersects the circle C at two points P and Q
If the line L1 is perpendicular to L and the line L1 intersects the circle C at two points m and N, the maximum area of the quadrilateral pmqn can be obtained

Let the equation of the circle be X & # 178; + Y & # 178; + DX + ey + F = 0, the center of the circle C on the straight line y = x, d = e, substituting points a (- 2,0), B (0,2) into 4-2d + F = 04 + 2D + F = 0, the solution is: F = - 4, d = e = 0, the equation of the circle is X & # 178; + Y & # 178; = 4, let the parameter equation of the straight line L: y = KX + 1 be {x = TCOS θ, y = 1 + Tsin θ (θ is tilt