Find a point on the surface z=8-(3*x^2)*y-(2*y^2) where the tangent plane is perpendicular to the line x=2-3t, y=7+8t and z=5-t.I don't understand how x=2-3t, y=7+8t and z=5-t makes a line?

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that a level surface is expressed by the function `f(x,y,z) =0`  and you need to find  at what point the tangent plane is perpendicular to the line given parametrically:

`x=2-3t`

`y=7+8t`

`z=5-t`

You should come up with the substitution for the tangency point (a,b,c).

Since the tangency point belongs to level surface `8 - (3*x^2)*y - (2*y^2) - z = 0` , then its coordinates need to check the equation such that:

`f(a,b,c) = 8 - (3*a^2)*b - (2*b^2) -c = 0 `

You need to remember that gradient vector of `f(x,y,z) `  is orthogonal to the level surface such that:

`grad` `f(x,y,z) =`  `((del f)/(del x),(del f)/(del y),(del f)/(del z))`

`grad f(x,y,z) = (6xy, -3x^2-4y,-1)`

`grad f(a,b,c) = (6ab, -3a^2-4b,-1)`

You should remember that gradient vector is also orthogonal to the tangent plane.

Notice that the problem provies the information that the tangent plane is orthogonal to the line `x=2-3t,y=7+8t,z=5-` t, hence the line `x=2-3t,y=7+8t,z=5-t`  is perpendicular to tangent plane.

You need to remember that two lines perpendicular to a plane are parallel, hence the line `x=2-3t,y=7+8t,z=5-t`  is parallel to gradient vector.

You need to express the vector of line `x=2-3t,y=7+8t,z=5-t ` such that `bar d = lt-3,8,-1gt` .

You need to express the condition that the vectors `bar d`  is parallel to vector `grad`  such that:

`(6ab)/(-3) = (-3a^2-4b)/8 = (-1)/(-1)`

`9a^2 + 12b = 48ab=gt 9a^2= 48ab -12b`

You need to divide by 3 such that:

`3a^2 = 16ab - 4b`

Factoring out b yields:

`3a^2 = b(16a - 4) =gt b = (3a^2)/(16a - 4)`

You need  to substitute `-1/(2a)`  for b in`-3a^2-4b = 8`  such that:

`-3a^2 + 2/a = 8`

You need to bring the terms to a common denominator such that:

`-3a^3 - 8a + 2 = 0 =gt 3a^3+ 8a- 2 = 0`

 

Notice that graph intercepts x axis at a value between 0 and 1, hence the equation  `3a^3 + 8a - 2 = 0`  has one real value over `(0,1).`

Hence, evaluating the point where the tangent plane to surface  `z=8-(3*x^2)*y-(2*y^2)`  is perpendicular to the line given parametrically yields `(a,-1/(2a),1),`  where `a in (0,1).`

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cosinusix | College Teacher | (Level 3) Assistant Educator

Posted on

This is a parametric equation of a line. The way it is constructed: each time t provides the coordinates of a position. The set is the set of all the possible positions for any real time. The equations are linear therefore it is a line. An other way to see that it is a line it to use the third equation to substitute t in the two first equations. t=5-z, therefore x=2-3(5-z)=3z-13, y=7+8(5-z)=47-8z. The set you are considering is included in x=3z-13 ( plane ) and y=47-8z (second plane). The set of point is therefore included in the intersection of the two planes which is a line.

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