what is the pt of tangency of the sphere to a parallel plane at the other end of the diameter & what is the?

Question

a sphere with center (7,1,-2) is tangent to a plane 2x+3y-z=5. what is the pt of tangency of the sphere to a parallel plane at the other end of the diameter & what is the rectangular equation of that tangent plane?

so the answer is (9,4,-3) and 2x+3y-z=33, but how do u get this answer?

Answer 1

A.) I can first help you out with the second part. [Later: in fact, logically, it must be answered first.]

The distance of a point (x1, y1, z1) from a plane

ax + by + cz + d = 0

is mod (a x1 + b y1 +c z1 + d) / sqrt(a^2 + b^2 + c^2).

So the distance of the point (7, 1, -2) [the sphere's centre] from the plane 2x + 3y - z - 5 = 0 is:

mod(2*7 + 3*1 -1*(-2) - 5) / sqrt(4 + 9 + 1) = mod(14) / sqrt(14) = sqrt(14).

Thus the radius of the sphere is sqrt(14).

But the centre must be the same distance from the other tangent plane, which since it's parallel to the first one must also have an equation of the form:

2x + 3y - z - C = 0

So we must also have

mod(2*7 + 3*1 -1*(-2) - C) / sqrt(14) = sqrt(14), so that

mod(19 - C) must be 14 once again. But C = 5 being already taken, the only other number that will satisfy this is C = 33, which gives mod(-14) i.e. mod(14), or 14, as before.

So the other plane is

2x + 3y - z = 33. ......(1) QED

[As Samuel Pepys would say: "And so to bed."}

B.) Back up with the lark:

The equation of the normal to the tangent planes through the centre, C (7, 1, - 2), is given by:

(x - 7)/2 = t, (y - 1)/3 = t, and (z + 2)/(- 1) = t, ......(2)

where t is a parameter that varies along that normal. Inserting those values for x, y and z along the normal into the equation for the tangent plane 2x + 3y - z = 33 (1), one finds:

2(7 + 2t) + 3 (1 + 3 t) - (- 2 - t) = 33, that is

19 + 4 t = 33, i.e. t = 1.

Putting that value of t back into the parameterized equation for the line (2), one finds that the intersection point is:

(9, 4, - 3). QED

And so we've put the entire problem to bed.

Live long and prosper.

Answer 2

Dr Spock is correct except for one little error on this statement near the end.

19 + 4 t = 33, i.e. t = 1

It should say "14 t".

19 + 14 t = 33, i.e. t = 1