(x- h)^2+(y- k)^2= r^2
The standard equation of a sphere with radius r and center ( h, k, l) is very similar to the standard equation of a circle.
(x- h)^2+(y- k)^2+(z- l)^2= r^2
In our case, the sphere has center ( 0, 0, 0), and a radius of 5.
(x- h)^2+(y- k)^2+(z- l)^2= r^2
⇓
(x- 0)^2+(y- 0)^2+(z- 0)^2= 5^2
⇕
x^2+y^2+z^2=25
Now, let us write the conditions for checking the relative position of a point and the sphere.
A point ( a, b, c) lies on the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2=25.
A point ( a, b, c) lies inside the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2<25.
A point ( a, b, c) lies outside the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2>25.
We have to check the position of point A( 0, - 3, 4) with respect to the sphere. Let's substitute the values into the equation of the sphere.
b From Part A we know that the given sphere has the equation x^2+y^2+z^2=25. Now, let's recall the conditions for checking the relative position of a point and the sphere.
A point ( a, b, c) lies on the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2=25.
A point ( a, b, c) lies inside the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2<25.
A point ( a, b, c) lies outside the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2>25.
We have to check the position of point B( 1, - 1, - 1) with respect to the sphere. Let's substitute the values into the equation of the sphere.
c From Part A we know that the given sphere has the equation x^2+y^2+z^2=25. Now, let's recall the conditions for checking the relative position of a point and the sphere.
A point ( a, b, c) lies on the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2=25.
A point ( a, b, c) lies inside the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2<25.
A point ( a, b, c) lies outside the sphere x^2+ y^2+ z^2=25 when a^2+ b^2+ c^2>25.
We have to check the position of point C( 4, - 6, - 10) with respect to the sphere. Let's substitute the values into the equation of the sphere.
