Turing

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question:the Cartesian coordinate system, it is known that the vertex A has coordinates (0， sqrt{2}) and vertex B has coordinates (0， sqrt{2}). If the product of the slopes of lines PA and PB is 2, then the equation of the trajectory of point P is ( ) A: frac {y^{2}}{2}+x^{2}=1 B: frac {y^{2}}{2}+x^{2}=1 (x≠0) C: frac {y^{2}}{2}x^{2}=1 D: frac {y^{2}}{2}+x^{2}=1 (y≠0)

answer:Let the coordinates of the moving point P be (x, y). Then, according to the given condition, we have the product of the slopes of lines PA and PB as frac {y+ sqrt {2}}{x} cdot frac {y- sqrt {2}}{x}=-2. By multiplying the fractions, we get: left(frac {y+ sqrt {2}}{x}right) left(frac {y- sqrt {2}}{x}right) = -1 Next, we expand the multiplication: frac {(y+ sqrt {2})(y- sqrt {2})}{x^2} = -2 Simplifying the left hand side by multiplying the binomials: frac {y^2 - (sqrt {2})^2}{x^2} = -2 We now substitute (sqrt {2})^2 by 2: frac {y^2 - 2}{x^2} = -2 In the next step, we isolate the term y^2 - 2: y^2 - 2 = -2x^2 Adding 2 to both sides gives us the trajectory equation for point P: y^2 = -2x^2 + 2 We notice that the value of x can't be 0 because it would make the original slopes undefined. So, we divide each side of the equation by 2 and rearrange it to get the desired form: frac {y^2}{2} + x^2 = 1 quad (x neq 0) Therefore, the trajectory of point P is frac {y^2}{2} + x^2 = 1 for x neq 0. Thus, the correct option is boxed{B}.

question:Consider a line that passes through the points (4, 10), (3, m), and (12, 5). Determine the value of m.

answer:For the points to lie on the same line, the slopes between successive points must be equal. Therefore, the slope between (4, 10) and (-3, m) should equal the slope between (-3, m) and (-12, 5). 1. **Calculate the first slope**: [ text{slope}_{1} = dfrac{m-10}{-3-4} = dfrac{m-10}{-7} ] 2. **Calculate the second slope**: [ text{slope}_{2} = dfrac{5-m}{-12-(-3)} = dfrac{5-m}{-9} ] 3. **Set the two slopes equal to each other**: [ dfrac{m-10}{-7} = dfrac{5-m}{-9} ] 4. **Solve for (m)**: [ -9(m-10) = -7(5-m) implies -9m + 90 = -35 + 7m implies -16m = -125 implies m = dfrac{125}{16} ] 5. **Final Answer**: [ m = boxed{dfrac{125}{16}} ]

question:find value for x from below equation ? x + 1 = 3

answer:To find the value of x, you need to isolate x on one side of the equation. You can do this by subtracting 1 from both sides of the equation: x + 1 - 1 = 3 - 1 This simplifies to: x = 2 So the value of x is boxed{2} .

question:a hexagon ABCDEF, where angle A is unknown, the following angles are given: angle B = 134^circ, angle C = 98^circ, angle D = 120^circ, angle E = 139^circ, angle F = 109^circ. What is the measure of angle A?

answer:The sum of the internal angles of a hexagon is calculated by the formula 180(n-2), where n is the number of sides. For a hexagon, n=6, so: [ 180(6-2)= 180 times 4 = 720^circ ] Now, using the given angle measures: [ angle A + 134^circ + 98^circ + 120^circ + 139^circ + 109^circ = 720^circ ] [ angle A + 600^circ = 720^circ ] Subtract 600^circ from both sides: [ angle A = 720^circ - 600^circ = 120^circ ] Thus, the measure of angle A is boxed{120^circ}.