Points C (1; -1.5; 3) and D (-1; 2.5; -3) lie on the sphere. The centers of the sphere belong to the segment CD.

| September 24, 2021 | education

Points C (1; -1.5; 3) and D (-1; 2.5; -3) lie on the sphere. The centers of the sphere belong to the segment CD. a) Write down the equation of the sphere b) Do the points with coordinates (3; -1.5; √7) and (1; 2.5; 3) belong to the sphere

As is known, the canonical equation of a sphere centered at the point O (x0; y0; z0) and of radius R has the form (x – x0) ^ 2 + (y – y0) ^ 2 + (z – z0) ^ 2 = R ^ 2 …
Since the points C (1; –1.5; 3) and D (–1; 2.5; –3) lie on the sphere and the center of the sphere belongs to the segment CD, it can be argued that the segment CD is the diameter of the sphere and the center of the sphere is in the middle of the CD segment.
In order to find the length of the diameter, we use the formula for calculating the distance between two points A (xa; ya; za) and B (xb; yb; zb): AB = √ [(xb – xa) ^ 2 + (yb – ya) ^ 2 + (zb – za) ^ 2]. We have СD = √ [(- 1 – 1) ^ 2 + (2.5 – (–1.5)) ^ 2 + (–3 – 3) ^ 2] = √ (2 ^ 2 + 4 ^ 2 + 6 ^ 2) = √ (4 + 16 + 36) = √ (56) = 2√ (14). Hence, R = СD: 2 = 2√ (14): 2 = √ (14).
Now we determine the coordinates of the center of the sphere O (x0; y0; z0). We have x0 = (xc + xd): 2 = (1 + (–1)): 2 = 0: 2 = 0; y0 = (yc + yd): 2 = (–1.5 + 2.5): 2 = 1: 2 = 0.5; z0 = (zc + zd): 2 = (3 + (–3)): 2 = 0: 2 = 0.
Thus, the desired equation has the form: (x – 0) 2 + (y – 0.5) 2 + (z – 0) 2 = 14 or x2 + (y – 0.5) 2 + z2 = 14.
Let us check that the points with coordinates (3; –1.5; √ (7)) and (1; 2.5; 3) belong to the sphere. We have 32 + (–1.5 – 0.5) ^ 2 + (√ (7)) ^ 2 = 9 + 16 + 7 = 32 ≠ 14, therefore, the point with coordinates (3; –1.5; √ ( 7)) does not belong to the sphere. Similarly, we have 12 + (2.5 – 0.5) ^ 2 + 3 ^ 2 = 1 + 4 + 9 = 14, therefore, the point with coordinates (1; 2.5; 3) belongs to the sphere.
Answers: x ^ 2 + (y – 0.5) ^ 2 + z ^ 2 = 14; the point with coordinates (3; –1.5; √ (7)) does not belong to the sphere; the point with coordinates (1; 2.5; 3) belongs to the sphere.



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