Question

Q4. (5 points). Find the equation of the plane that passes through the line of intersection of the two planes x - 2y = 3 and y- z = 0 and parallel to the line x = y - 1 = 2+1 Q5. (4 points). Find the distance from the point A(1,2,3) and the line 2+1 y-1 2 Q6. (4 points). Give the name and sketch the surface whose equation is given by x2 + 2y2 – 12y – z = 0

Answer 1

and y=z the planes Given planes are x-2y = 3 intersection :- line passing through both y = Z = (from plane 1 24 = x-3 y = x-2) x-3 2-3 Z MO y 1 -- Required plane A Hence vector a long the above line will be 1a-2 itt y-1 zt Also the plane is parallel to line x2 T line ! b the above î+ / +2 Hence vector b along unit to → Let û be the normal the plane ! << Then ã xi lallbl -. (2 i + ) tk) x (it) + 2%) Jo 2 ! 2 - = + (î-3 +) Hence the plane will be axt by otcz =d where ao!, b=-3,C? | x - 3y + z = =d Intersection point of line 6 & line ② on the plane: - lies Let x - 2 = Ž= z.. Z+1 & & II = 24 & x=C2 y = (2+1 • Z=2₂-1 x = 20, +3 , yale = 1, comparing both sides 26, +3=C2 C = C₂+1 soling the above equations ? C = - 4 C2

Hence the point on the plane is (20, +3, c,,G) Putting this point on the equation of plane x-3y + zad -s + 12 - 4 d d=3 Hence the equation of the plane is x-3y +z=3 A(1,2,3) . - Any point on the line x=4-1 = Zt1 written as we have written in the previone page (C2, C2+1, 2C₂-1). can be ztl Ray-1 = BCC₂, C2+1, 26-1) Vector along the direction of the line D = + 2 vector (1-(2) it (2 - (C2+1) ŷ + -(3-(2(2-1)) Bca) i + (1-C2) Î + (2 – 2 (2) Ř I and An iu go, I. AB=0 Since angle between - O = (1-C2) + (1-12) + 2x ( 2-2 (2) 2 - 2 ₂ + 4 - 4C₂ 2 = 1 Hence coordinates of point B=(1,2,1). between line - Hence distance AB and paint A √(1-1)² + (2-2)2 + (3-1)2 2.

Q6: Giren equation of surface x² + 2y² - 12y-Z = 0 x ² + 2 (y 2 - 64) 2 ( 4² – 64 +9) 2 (y - 3)2 2 + - 18 - Z 1 + Z + 18 >> Z +18 1 + (12) Hence this is an elliptic parabollood. Contatus of the parabellion z x=0, y =3 → Z=-18 Top view -18 on x-y planeta 0,640) 16,20) (V18,0,0) Side view on x-z plane (-Vī8,0,0) Z side view on yoz plane Vis JIS Z -18 •Y O