8.05g Tangent planes: equation at a given point on surface

2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).

1. Find a normal vector at the point \(( x , y , z )\) on the surface.
2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.

2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
2. Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
3. The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
5. Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).

2 You are given that \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 2 y ^ { 2 } - z ^ { 2 } + 2 x z + 2 y z + 4 z - 3\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). The surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), and \(\mathrm { P } ( - 2 , - 1,1 )\) is a point on \(S\).
2. Find an equation for the normal line to the surface \(S\) at the point P .
3. A point Q lies on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find the constant \(c\) such that \(\mathrm { PQ } \approx c | h |\).
4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
5. Given that \(x + y + z = k\) is a tangent plane to the surface \(S\), find the two possible values of \(k\).

2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Find the equation of the normal line to the surface \(S\) at the point P .
3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.

2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the three stationary points on the surface.
3. Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
4. The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
5. Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).

2 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x\).

1. Sketch the section of \(S\) given by \(y = - 3\), and sketch the section of \(S\) given by \(x = - 6\). Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
2. From your sketches in part (i), deduce that \(( - 6 , - 3 , - 324 )\) is a stationary point on \(S\), and state the nature of this stationary point.
3. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\), and hence find the coordinates of the other three stationary points on \(S\).
4. Show that there are exactly two values of \(k\) for which the plane with equation $$120 x - z = k$$ is a tangent plane to \(S\), and find these values of \(k\).

2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.

2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).

1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).

1 The surface \(E\) has equation \(z = \sqrt { 500 - 3 x ^ { 2 } - 2 y ^ { 2 } }\).

1. Determine the values of \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\) at the point \(P\) on \(E\) with coordinates \(( 11 , - 8,3 )\).
2. Find the equation of the tangent plane to \(E\) at \(P\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\) where \(a , b , c\) and \(d\) are integers.

2 A surface \(S\) has equation \(\mathrm { z } = 4 \mathrm { x } \sqrt { \mathrm { y } } - \mathrm { y } \sqrt { \mathrm { x } } + \mathrm { y } ^ { 2 }\) for \(x , y \geqslant 0\). Determine the equation of the tangent plane to \(S\) at the point (1,4,20). Give your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\) where \(a , b , c\) and \(d\) are integers.

6 A surface \(S\) has equation \(z = \mathrm { f } ( x , y )\), where \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y\). It is given that \(S\) has a single stationary point, \(P\).

1. (a) Determine the coordinates of \(P\).
   (b) Determine the nature of \(P\).
2. Find the equation of the tangent plane to \(S\) at the point \(Q ( 1,2,38 )\).

5 A surface \(S\) is defined by \(z = f ( x , y )\), where \(f ( x , y ) = y e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }\).

1. 1. Find \(\frac { \partial f } { \partial x }\).
   2. Show that \(\frac { \partial f } { \partial y } = - \left( x ^ { 2 } y + 2 x y + 2 y - 1 \right) e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }\).
   3. Determine the coordinates of any stationary points on \(S\). Fig. 5.1 shows the graph of \(z = e ^ { - x ^ { 2 } }\) and Fig. 5.2 shows the contour of \(S\) defined by \(z = 0.25\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_686_822_244} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_437_822_1105} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
      \end{figure}
2. Specify a sequence of transformations which transforms the graph of \(\mathrm { z } = \mathrm { e } ^ { - \mathrm { x } ^ { 2 } }\) onto the graph of the section defined by \(z = f ( x , 1 )\).
3. Hence, or otherwise, sketch the section defined by \(z = f ( x , 1 )\).
4. Using Fig. 5.2 and your answer to part (c), classify any stationary points on \(S\), justifying your answer. You are given that \(P\) is a point on \(S\) where \(z = 0\).
5. Find, in vector form, the equation of the tangent plane to \(S\) at \(P\). The tangent plane found in part (e) intersects \(S\) in a straight line, \(L\).
6. Write down, in vector form, the equation of \(L\).

3 A surface, \(S\), is defined by \(g ( x , y , z ) = 0\) where \(g ( x , y , z ) = 2 x ^ { 3 } - x ^ { 2 } y + 2 x y ^ { 2 } + 27 z\). The normal to \(S\) at the point \(\left( 1,1 , - \frac { 1 } { 9 } \right)\) and the tangent plane to \(S\) at the point \(( 3,3 , - 3 )\) intersect at \(P\). Determine the position vector of P .

1 A surface, \(S\), is defined in 3-D by \(z = f ( x , y )\) where \(f ( x , y ) = 12 x - 30 y + 6 x y\).

1. Determine the coordinates of any stationary points on the surface.
2. The equation \(\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { a } )\), where \(a\) is a constant, defines a section of S . Given that this equation is \(\mathrm { z } = 24 \mathrm { x } + \mathrm { b }\), find the value of \(a\) and the value of \(b\). The diagram shows the contour \(z = 12\) and its associated asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{33c9e321-6044-45c4-bf37-0a6da3ecaf0d-2_860_1143_742_242}
3. Find the equations of the asymptotes.
4. By forming grad \(g\), where \(g ( x , y , z ) = f ( x , y ) - z\), find the equation of the tangent plane to \(S\) at the point where \(x = 3\) and \(y = 2\). Give your answer in vector form. The point \(( 0,4 , - 120 )\), which lies on S , is denoted by A .
   The plane with equation \(\mathbf { r }\). \(\left( \begin{array} { r } 3 \\ 3 \\ - 2 \end{array} \right) = 52\) is denoted by \(\Pi\).
5. Show that the normal to S at A intersects \(\Pi\) at the point \(( - 360,304 , - 110 )\).

6 A surface \(S\) is defined by \(z = \mathrm { f } ( x , y ) = 4 x ^ { 4 } + 4 y ^ { 4 } - 17 x ^ { 2 } y ^ { 2 }\).

1. 1. Show that there is only one stationary point on \(S\). The value of \(z\) at the stationary point is denoted by \(s\).
   2. State the value of \(s\).
   3. By factorising \(\mathrm { f } ( x , y )\), sketch the contour lines of the surface for \(z = s\).
   4. Hence explain whether the stationary point is a maximum point, a minimum point or a saddle point. C is a point on \(S\) with coordinates ( \(a , a , \mathrm { f } ( a , a )\) ) where \(a\) is a constant and \(a \neq 0\). \(\Pi\) is the tangent plane to \(S\) at C .
2. 1. Find the equation of \(\Pi\) in the form r.n \(= p\).
   2. The shortest distance from the origin to \(\Pi\) is denoted by \(d\). Show that \(\frac { d } { a } \rightarrow \frac { 3 \sqrt { 2 } } { 4 }\) as \(a \rightarrow \infty\).
   3. Explain whether the origin lies above or below \(\Pi\). \section*{END OF QUESTION PAPER}

3 The surface \(S\) has equation \(z = \frac { x } { y } \sin y + \frac { y } { x } \cos x\) where \(0 < x \leqslant \pi\) and \(0 < y \leqslant \pi\).

1. Find
   • \(\frac { \partial z } { \partial x }\),
   • \(\frac { \partial z } { \partial y }\).
   • Determine the equation of the tangent plane to \(S\) at the point \(A\) where \(x = y = \frac { 1 } { 4 } \pi\). Give your answer in the form \(a x + b y + c z = d\) where \(a , b , c\) and \(d\) are exact constants.
   • Write down a normal vector to \(S\) at \(A\).

1 A surface has equation \(z = x \tan y\) for \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\).

1. Find

2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).

1. Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
2. A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
   (A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
   (B) Deduce that the stationary point A is a local minimum.
   (C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B .
3. The point Q with coordinates \(( 1,1,53 )\) lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
4. The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\). Find the coordinates of R .

A surface \(S\) is defined for \(z \geqslant 0\) by \(x^2 + y^2 + 2z^2 = 126\). \(C\) is the set of points on \(S\) for which the tangent plane to \(S\) at that point intersects the \(x\)-\(y\) plane at an angle of \(\frac{1}{4}\pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined. [6]

A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).

1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]

The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).

1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]

A surface \(S\) has equation \(z = f(x, y)\), where \(f(x, y) = 2x^2 - y^2 + 3xy + 17y\). It is given that \(S\) has a single stationary point, \(P\).

1. Determine the coordinates, and the nature, of \(P\). [8]
2. Find the equation of the tangent plane to \(S\) at the point \(Q(1, 2, 38)\). [2]