1 Let A3 50 and B1 32 be two points and let S be a sphere wh

1. Let A(3, -5,0) and B(-1, 3,2) be two points and let S be a sphere which is centered at the midpoint C of the line segment .4B and passing through point D(2, 3, -2). Find au equation of sphere S. 2. Find the vector of norm 20 in the direction opposite of vector 3i - 2j + square root 3k 3. Find the value(s) of r such that a = -2i+5j+ik is two times as long as b = 3i+j-k.

Solution

1) Ans- mid point of ab= 1,-1,1

therefore (x-1)² + (y+1)² + (z-1)² = r²

now, 2,3,-2 satisfies, therefore, 1 + 16 + 9 = r² = 26

equation is (x-1)² + (y+1)² + (z-1)² = 26

2) direction opp means just flip the signs of i,j,k and divide by its magnitude, now multiply by 20

= 20*((-3i + 2j - 3^0.5 k ) / 4) = -15i +10j - 5*3^0.5k

3) twice means magnitude of A vector is two times B vector

|A|= sqrt( 4 + 25 + x²) = 2*sqrt( 9 + 1 + 1)

29 +x² = 44

x= +-sqrt(15)