Normal vector or normal line to surface

A question is this type if and only if it asks to find a normal vector to a surface at a point or the equation of the normal line.

3 questions · Challenging +1.2

8.05d Partial differentiation: first and second order, mixed derivatives8.05f Nature of stationary points: classify using Hessian matrix
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OCR MEI FP3 2006 June Q2
24 marks Challenging +1.2
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.
OCR MEI FP3 2012 June Q2
24 marks Challenging +1.2
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 \(k\).
OCR MEI FP3 2014 June Q2
24 marks Challenging +1.2
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.