Find the points on z-axis which are at a distance √21 from the poin

1.	Find the points on z-axis which are at a distance √21 from the point (1, 2, 3).
Answer» Given: Points A(1, 2, 3) To find: the point on z-axis which is at distance of from the given point As we know x = 0 and y = 0 on z-axis Let R(0, 0, z) any point on z-axis According to question: RA = \(\sqrt{21}\) ⇒ RA² = 21 Formula used: The distance between any two points (a, b, c) and (m, n, o) is given by, \(\sqrt{(a−m)^2+(b−n)^2+(c−0)^2}\) Therefore, Distance between R(0, 0, z) and A(1, 2, 3) is RA, = \(\sqrt{(0−1)^2+(0−2)^2+(z−3)^2} \) = \(\sqrt{1+4+(z−3)^2} \) = \(\sqrt{5+(z−3)^2} \) As RA² = 21 5 + (z – 3)² = 21 ⇒ z²+ 9 – 6z + 5 = 21 ⇒ z² – 6z = 21 – 14 ⇒ z²– 6z – 7 = 0 ⇒ z²– 7z + z – 7 = 0 ⇒ z(z– 7) + 1(z – 7) = 0 ⇒ (z– 7) (z + 1) = 0 ⇒ (z– 7) = 0 or (z + 1) = 0 ⇒ z= 7 or z = -1 Hence points (0, 0, 7) and (0, 0, -1) on z-axis is equidistant from (1, 2, 3)

Find the points on z-axis which are at a distance √21 from the point (1, 2, 3).

Answer»

Given: Points A(1, 2, 3)

To find: the point on z-axis which is at distance of from the given point

As we know x = 0 and y = 0 on z-axis

Let R(0, 0, z) any point on z-axis

According to question:

RA = \(\sqrt{21}\)

⇒ RA² = 21

Formula used:

The distance between any two points (a, b, c) and (m, n, o) is given by,

\(\sqrt{(a−m)^2+(b−n)^2+(c−0)^2}\)

Therefore,

Distance between R(0, 0, z) and A(1, 2, 3) is RA,

= \(\sqrt{(0−1)^2+(0−2)^2+(z−3)^2} \)

= \(\sqrt{1+4+(z−3)^2} \)

= \(\sqrt{5+(z−3)^2} \)

As RA² = 21

5 + (z – 3)² = 21

⇒ z²+ 9 – 6z + 5 = 21

⇒ z² – 6z = 21 – 14

⇒ z²– 6z – 7 = 0

⇒ z²– 7z + z – 7 = 0

⇒ z(z– 7) + 1(z – 7) = 0

⇒ (z– 7) (z + 1) = 0

⇒ (z– 7) = 0 or (z + 1) = 0

⇒ z= 7 or z = -1

Hence points (0, 0, 7) and (0, 0, -1) on z-axis is equidistant from (1, 2, 3)