3D Geometry Test

Result

3D Geometry Test - 2

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Question 1
1 / -0

Centre of the sphere which touches the lines y = x , z = c and y = - x , z = - c lies on

A
xy + 2cz = 0

B
None of these

C
yz + 2cx = 0

D
zx + 2cy = 0

Solution

Let equation of sphere be

x² + y² + z² + 2ux + 2vy + 2wz + d = 0

The line y = x , z = c intersects it at the point where

2x² + 2 ( u + v ) x + ( c² + 2wc + d ) = 0

Since the line touches the sphere, the roots of this equation are coincident.

∴ 4 ( u + v )² - 8 ( c² - 2wc + d ) = 0 ....... (1)

Similarly for second line

4 ( u - v )² - 8 ( c² - 2wc + d ) = 0 ...... (2)

Subtracting (2) from (1), we get

4uv - 8wc = 0

uv = 2wc.

∴ Locus of centre ( - u, - v, - w ) is xy = - 2cz.