The coordinates of a point which is equidista | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the point be $P(x, y, z)$. Since $P$ is equidistant from the given points $O(0, 0, 0), A(a, 0, 0), B(0, b, 0)$,and $C(0, 0, c)$,we have $PO^2

Vedclass · Accepted Answer

$\left( \frac{a}{2}, \frac{b}{2}, \frac{c}{2} \right)$

Vedclass · Answer

(A) Let the point be $P(x, y, z)$. Since $P$ is equidistant from the given points $O(0, 0, 0), A(a, 0, 0), B(0, b, 0)$,and $C(0, 0, c)$,we have $PO^2 = PA^2 = PB^2 = PC^2$.$x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2 = x^2 + (y - b)^2 + z^2 = x^2 + y^2 + (z - c)^2$.From $x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2$,we get $x^2 = x^2 - 2ax + a^2$,which implies $2ax = a^2$,so $x = \frac{a}{2}$.Similarly,from $x^2 + y^2 + z^2 = x^2 + (y - b)^2 + z^2$,we get $y = \frac{b}{2}$.And from $x^2 + y^2 + z^2 = x^2 + y^2 + (z - c)^2$,we get $z = \frac{c}{2}$.Thus,the coordinates of the point are $\left( \frac{a}{2}, \frac{b}{2}, \frac{c}{2} \right)$.

The coordinates of a point which is equidistant from the points $(0, 0, 0), (a, 0, 0), (0, b, 0)$ and $(0, 0, c)$ are given by

Similar Questions

The coordinates of the point $P$ which is equidistant from the points $A(a, 0, 0)$,$B(0, b, 0)$,$C(0, 0, c)$,and $O(0, 0, 0)$ are .......... where $a, b, c \neq 0$.

Let $A, B, C$ be the feet of perpendiculars from a point $P$ on the $x, y, z$-axes respectively. Find the coordinates of $A, B$ and $C$ for the following points $P$:
$(i)$ $(3, 4, 2)$
$(ii)$ $(-5, 3, 7)$
$(iii)$ $(4, -3, -5)$

Let $A, B, C$ be the feet of the perpendiculars from a point $P$ on the $xy, yz,$ and $zx$-planes respectively. Find the coordinates of $A, B, C$ for the following points $P$:
$(3, 4, 5), (-5, 3, 7), (4, -3, -5)$

Find the coordinates of a point on the $y$-axis which is at a distance of $5\sqrt{2}$ from the point $P(3, -2, 5)$.

If $A(1, 2, 3)$ and $B(-1, -1, -1)$ are two points,then the distance $AB$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module