Question

8. Find the equation of a sphere which passes through the origin and
(1) makes equal intercepts of unit length on the axes.
(ii) makes intercepts 3, 4, 5 from the co-ordinate axes.​

Answer 1

Since the circle cuts intercepts of a and b on the x axis, y axis and z axis respectivley.

.

Answer 2

Answer:

(1) The equation of sphere is $(x-1)^2+(y-1)^2+(z-1)^2=r^2$.

(2) The equation of sphere is $(x-3)^2+(y-4)^2+(z-5)^2=r^2$.

Step-by-step explanation:

Equation of sphere is,

$(x-h)^2+(y-k)^2+(z-l)^2=r^2$,

where $h,k,l$ are the intercepts on the $x$ - axis, $y$ - axis and  $z$ - axis respectively.

(1) According to the question,

The sphere makes equal intercepts of unit length on the $x$ - axis, $y$ - axis and  $z$ - axis respectively, i.e.,

$(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ on the $x$ - axis, $y$ - axis and  $z$ - axis respectively.

Let the radius of sphere be $r$.

Then the equation of sphere is,

$(x-1)^2+(y-1)^2+(z-1)^2=r^2$.

(2) According to the question,

The sphere makes intercepts of $3$, $4$ and $5$ on the $x$ - axis, $y$ - axis and  $z$ - axis respectively, i.e.,

$(3,0,0)$, $(0,4,0)$ and $(0,0,5)$ on the $x$ - axis, $y$ - axis and  $z$ - axis respectively.

Let the radius of sphere be $r$.

Then the equation of sphere is,

$(x-3)^2+(y-4)^2+(z-5)^2=r^2$.

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