The midppoint of the line segment from P_1 to P_2 is (-5,4). If P_1=(-2,2), what is P_2?
For the given equation list the intercepts and test for symmetry.
49x^2+y^2=49
Please show work
To find the other endpoint P2, we can use the midpoint formula, which states that the midpoint of a line segment is the average of its endpoints.
Given P1 = (-2, 2) and the midpoint (-5, 4), we can use the formula:
(x1 + x2)/2 = -5 and (y1 + y2)/2 = 4
Substituting the values for P1, we get:
((-2 + x2)/2) = -5 and ((2 + y2)/2) = 4
Solving for x2 and y2 separately:
(-2 + x2)/2 = -5 => -2 + x2 = -10 => x2 = -8
((2 + y2)/2) = 4 => 2 + y2 = 8 => y2 = 6
Therefore, P2 = (-8, 6).
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For the equation 49x^2 + y^2 = 49, let's find the x-intercepts, y-intercepts, and test for symmetry.
To find the x-intercepts, set y = 0 and solve for x:
49x^2 + 0^2 = 49
49x^2 = 49
x^2 = 1
x = ±1
So, the x-intercepts are (-1, 0) and (1, 0).
To find the y-intercepts, set x = 0 and solve for y:
49(0)^2 + y^2 = 49
0 + y^2 = 49
y^2 = 49
y = ±7
So, the y-intercepts are (0, -7) and (0, 7).
To test for symmetry, there are two types: symmetry with respect to the y-axis (also called the origin) and symmetry with respect to the x-axis.
- For symmetry with respect to the y-axis, replace x with -x in the equation:
49(-x)^2 + y^2 = 49
49x^2 + y^2 = 49
The equation remains the same, so the graph is symmetrical with respect to the y-axis (origin).
- For symmetry with respect to the x-axis, replace y with -y in the equation:
49x^2 + (-y)^2 = 49
49x^2 + y^2 = 49
Again, the equation remains the same, so the graph is symmetrical with respect to the x-axis.
In summary, the x-intercepts are (-1, 0) and (1, 0), the y-intercepts are (0, -7) and (0, 7), and the graph is symmetrical with respect to both the x-axis and y-axis (origin).