7. Match the following equations to the quadric surfaces shown. (One equation will be left unmatched.) 1. x2 = 322 + 2y2 4 Z 2. x2 = 322 - 2y2 3. x2 + 1 = 322+2y? 4. z + 1 =3x2+2y2 (a) (b) (c) 5. 22 - 1 = 3x2+2y2 6. -22 + 1 = 3x2 +232 7. z + 1 =3x2 -2y2 8. z + 1 =-3x2 +2y2 9. 1 = 322 - 2y2 (d) (e ) x (f ) 10. 1 = 322 + 2y2 7 (8) (h) (i) 8. Suppose that you have a point P1 = (1, y1, Z1), and you want to find the distance from P1 to the plane ax + by + cz + d = 0. Pick,,gi , Z1 ) (a) What is the most natural choice for n, the normal vector for the plane? D (b) Suppose that Po = (To, yo, Zo) is an arbitrary point in the plane. Write the vector from Po to P1 as b = PoPi in terms of To, I1, etc.. (c) The distance D from P1 to the plane is then just Po the absolute value of the scalar projection of b onto n. Accordingly, use formulas from 12.3 to write D in terms of b and n. (d) Now, using To, T1, a, etc., you should be able to obtain D = lax1 + by1 + cz1 + d| Va2 + 62 + c 2 (e) Use your formula to find the distance from the point (1, -2, 4) to the plane 3x + 2y + 6z = 5. 9. (a) Show that the line r = 2 - t, y = 1+ 2t, z = 3 + t intersects the line x = 3+ 2s, y = -3s, z = 5 + s, and find the point of intersection. (b) Find the equation of the plane containing both of these lines