(Solved): answer all parts of the question and show all workings 4. Let \( S \) be the surface in \( \mathbb{ ...

answer all parts of the question and show all workings

4. Let \( S \) be the surface in \( \mathbb{R}^{3} \) defined by \( x^{2}-y^{2}+2 z^{2}=9 \). (a) Write down the equation of the tangent plane \( P \) to \( S \) at \( (1,0,-2) \). (b) Does the point \( (2,1,3) \) lie in the tangent plane \( P \) ? Justify your answer. (c) For what value of the real number \( t \) does the point \( (2-t, 1-t, 3+t) \) lie in \( P \) ? (d) Write down the parametric description of the normal line to \( S \) at \( (1,0,-2) \).