Find the rectangular equation of hte tangent plane at the opposite end of the diameter.?

Question

Given that C(2,-3,1) is the center of a sphere & T(9,1,-4) is the pt of tangency of the sphere to a plane.
Find the rectangular equation of hte tangent plane at the opposite end of the diameter.

Answer 1

The vector CT is (7, 4, -5), so the opposite end of the same diameter will be P = (2, -3, 1) - (7, 4, -5) = (-5, -7, 6).
CT is normal to the tangent plane, so the plane has equation
7x + 4y - 5z + d = 0
Put in the coordinates of P to get
7(-5) + 4(-7) - 5(6) + d = 0
<=> d = 93
So the equation is
7x + 4y - 5z + 93 = 0.

Answer 2

Given that C(2,-3,1) is the center of a sphere and T(9,1,-4) is the point of tangency of the sphere to a plane. Find the rectangular equation of the tangent plane at the opposite end of the diameter.

First, find the equation of the line r, thru C and T.

Define vector v.

v = TC = = <2-9, -3-1, 1-(-4)> = <-7, -4, 5>

Define line r(t).

r(t) = OC + tv = <2, -3, 1> + t<-7, -4, 5>
r(t) = <2 - 7t, -3 - 4t, 1 + 5t>

Now find the point P on the opposite end of the diameter. P is the distance || v || from C in the direction of v. Plug in t = 1 to find P.

r(1) = (2 - 7, -3 - 4, 1 + 5) = P(-5, -7, 6)

Now we have a point P in the desired plane and the normal vector v to the plane. We can now write the equation of the plane.

-7(x + 5) - 4(y + 7) + 5(z - 6) = 0
-7x - 35 -4y - 28 + 5z - 30 = 0
-7x - 4y + 5z - 93 = 0

Multiply thru by -1.

7x + 4y - 5z + 93 = 0

Answer 3

First rearrange the function to the type g(x,y,z) = 0 f(x,y) = y/x = z g(x,y,z) = y/x - z = 0 next compute the gradient to discover a classic to the exterior grad(g) = = <-y/x², a million/x, -a million> evaluate at P(a million,2,2) (notice: this factor is on the exterior) n = <-y/x², a million/x, -a million> = <-2, a million, -a million> enable r = be a place vector for a factor interior the tangent airplane. Then r-P is a vector interior the airplane. with the help of definition, "n" is customary to all vectors interior the tangent airplane, so n•(r-P) = 0 n•r = n•P -2x + y - z = (-2)(a million) + (a million)(2) + (-a million)(2) And the equation of the tangent airplane is -2x + y - z = -2