CAPTER 3 Cylinder 3.. DENTON A cylinder is a surface generated by a straight line which is parallel to a fixed line and intersects a given curve or touches a given surface. The fixed line is called the axis and the given curve is called the guiding curve of the cylinder. Any line on the surface of a cylinder is called its generator. 3.. EQUATON O A CYLNDER (a) To find the equation of a cylinder whose generators intersect the conic ax + hxy + by + gx + fy + c = 0, z = 0...() and are parallel to the line x/l = y/m = z/n....() Suppose P (x, y, z ) be any point on the cylinder. The equation of the generator through the point P and parallel to equation () are x x y y z z = =...(3) l m n The generator (3) meets the plane z = 0 in the point lz x n y mz,, 0 n K Since the generator (3) meets the given conic (), lz a x n K + h lz mz x y n K n K + b mz y n K + g lz x n K mz + f y n K + c = 0 or a (nx lz ) + h (nx lz ) (ny mz ) + b (ny mz ) + gn (nx lz ) + fn (ny mz ) + cn = 0 80
CYLNDER 8 Thus the locus of (x, y, z ) is a (nx lz) + h (nx lz) (ny mz) + b (ny mz) + gn (nx lz) + fn (ny mz) + cn = 0 This is the required equation of the cylinder. Corollary : The equation of the cylinder whose generators are parallel to the z-axis, then (l = 0, m = 0, n = 0) put in equation (4), we get ax + by + hxy + gx + fy + c = 0 Corollary : The equation of the form f(x, y) = 0 represents a cylinder whose generator are parallel to z-axis. Corollary 3: The equation of the cylinder whose axis is z-axis and whose generators intersection the circle x + y = a, z = 0, is given by x + y = a. 3.3. RGT CRCULAR CYLNDER A right circular cylinder is a surface generated by a straight line passing through the point on a fixed circle and is perpendicular to its plane. The normal to the plane of the circle through its centre is called the axis of the cylinder and the section by a plane which is perpendicular to the axis is called the normal section i.e., a circle. The radius of the normal section is also called the radius of the cylinder. The length of the perpendicular from any point on a right circular cylinder to its axis is equal to its radius. 3.4. EQUATON O A RGT CRCULAR CYLNDER To find the equation of a right circular cylinder whose axis is the line x x y y z z = = and whose radius is r. l m n Suppose P (x, y, z) be any point on the cylinder and PN be the length of the point perpendicular from the point P on a given line and the given line passes through the point A (x, y, z ). Let PN = r M r N P (x, y, z) A(x,y,z) AN = ( x x ) l + ( y y ) m + ( z z ) n l + m + n AP = distance between the point A and P = ( x x ) + ( y y ) + ( z z )
8 ENGNEERNG MATEMATCS Now, we have PN = AP AN = {(x x ) + (y y ) + (z z ) {( ) ( ) ( )} } l x x + m y y + n z z l + m + n ence PN = p = {(x x ) + (y y ) + (z z ) {( ) ( ) ( )} } l x x + m y y + n z z l + m + n Since, the radius of the given cylinder is r, so by definition of right circular cylinder we have p = r. i.e., r (l + m + n ) = {(x x ) + (y y ) + (z z ) } (λ + m + n ) {l(x x ) + m (y y ) + n (z z )} or r (l + m + n ) = [(y y ) n (z z ) m] + [(z z ) l (x x ) n] + [(x x ) m (y y ) l] or r (l + m + n ) = Σ[(y y )n (z z )m] This is required equation of right circular cylinder. 3.5. ENVELOPNG CYLNDER A cylinder whose generator touches a given surface and is directed in a given direction is called an enveloping cylinder. 3.6. EQUATON O AN ENVELOPNG CYLNDER To find the equation of the enveloping cylinder whose generator touch the sphere ax + by + cz =, and are parallel to line x y l = z m = n. Let P (x, y, z ) be any point on the given enveloping cylinder. The equation of the generator of the cylinder through the point P and parallel to the line x l = y z m = n is x x y y z z = = = r (say)...() l m n The coordinates if any point on the generator () are (lr + x, mr + y, nr + z ) Suppose the given sphere meets the point (lr + x, mr + y, nr + z ), then we have a (lr + x ) + b (mr + y ) + c (nr + z ) = or r (al + bm + cn ) + r (alx + bmy + cnz ) + ax + by + cz =...() The line () will touch given sphere if the equation () has equal roots. Therefore we get (alx + bmy + cnz ) = (al + bm + cn ) (ax + by + cz )
CYLNDER 83 ence the locus of (x, y, z ) is (alx + bmy + cnz ) = (al + bm + cn ) (ax + by + cz ) This is required equation of enveloping cylinder. 3.7. EQUATON O A TANGENT PLANE TO TE CYLNDER To find the equation of a tangent plane to the cylinder whose equation is ax + hxy + by + gx + fy + c = 0 at the point P (x, y, z ). Suppose the given equation of the cylinder is ax + hxy + by + gx + fy + c = 0...() Since, the point P (x, y, z ) line on (), then ax + hx y + by + gx + fy + c = 0...() Let the equation of a line which passes through the point P (x, y, z ) and whose direction x x y y z z cosine are l, m, n be = = = r...(3) l m n any point on the line (3) is (lr + x, mr + y, nr + z ) and the given cylinder () are given by a (lr + x ) + h (lr + x ) (mr + y ) + b (mr + y ) + g (lr + x ) + f (mr + y ) + c = 0 r (al + hlm + bm ) + r[l(ax + hy + g) + m (hx + by + f)] + (ax + hx y + by + gx + fy + c) = 0 Using equation (), we get r (al + hlm + bm ) + r [l(ax + hy + g) + m(hx + by + f)] = 0...(4) One root of this equation is zero. This given line (3) will be a tangent line at (x, y, z ). f the other root is also zero. Equation (4) other root is zero if l [hx + by + g] + m [hx + by + f}] = 0...(5) Eliminating l, m, n between equation (3) and (5), we get (x x )[hx + by + g] + (y y )[hx + by + f] = 0 or x (ax + hy + g) + y (hx + by + f) + gx + fy + c = ax + hx y + by + gx + fy + c = 0 Using equation (), we get x (ax + hy + g) + y (hx + by + f) + gx + fy + c = 0 This is the required equation of tangent plane to the cylinder.
84 ENGNEERNG MATEMATCS Corollary : The tangent plane at the point (x, y, z ) to the cylinder ax + hx y + by + gx + fy + c = 0...() x (ax + hy + g) + y (hx + by + f) + (gx + fy + c) = 0 or axx + h (xy + x y) + byy + g (x + x ) + f (y + y ) + c = 0 this equation is obtained by replacing x by xx, y by yy, x by x + x, y by y + y and xy by xy + yx in (). SOLVED EXAMPLES Example. ind the equation of a cylinder whose generators are parallel to the line x = y/ = z and passing through the curve is 3x + y =, z = 0. Sol. The equation of the giving curve is 3x + y =, z = 0...() The equation of the giving line is x y z = =...() Let us consider a point P (x, y, z ) on the cylinder. The equation of generator through the point P (x, y, z ) which is a line parallel to the given line () are x x y y z z = =...(3) The generator (3) meets the plane z = 0 in the point given by x x y y z z = = i.e., (x + z, y + z, 0) Since the generator (3) meets the conic (). ence the point (x + z, y + z, 0) will satisfy the equation of the conic given by (), we have 3 (x + z ) + (y + z ) = or 3 (x + x z + z ) + ( y + 4y z + 4z ) = x + 6x z + z + y + 8y z = 0 The locus of P (x, y, z ) is 3x + 6xz + z + y + 8yz = 0 This is required equation of the cylinder. Example. ind the equation of the circular cylinder whose generating lines have the direction cosines l, m, n and which pass through the fixed circle x + y = a in ZOX plane. Sol. The equation of the guiding curve (circle) are x + y = a, ZOX lane i.e., y = 0...()
CYLNDER 85 let us consider a point P (x, y, z ) on the cylinder. The equation of generator through the point P (x, y, z ) and with direction cosine l, m, n are x x y y z z = =...() l m n the generator () meet the plane y = 0 in point given by x x l y y z z = = m n i.e. x ly, 0, z m Since, the generator () meets the curve (). ence, the point x the equation of the curve given by (), we have ly x m K + ny z m K = a or (mx ly ) + (mz ny ) = a m The locus of P (x, y, z ) is (mx ly) + (mz ny ) = a m ny m K K ly ny, 0, z will satisfy m m This is the required equation of cylinder. Example 3. ind the equation of the cylinder which intersects the curve ax + by + cz =, lx + my + nz = p and whose generators are parallel to x-axis. Sol. The given equation of the guiding curve are ax + by + cz =...() and lx + my + nz = p...() Since, the generators of the cylinder are parallel to x-axis, so the equation of the cylinder will not contain terms of x. Thus the equation of the cylinder will be obtained by eliminates x between equation() and (), we get a l (p my nz) + by + cz = a(p my nz) + bl y + cl z = l or a (am + bl )y + (an + cl )z + amnyz amby anbz + (ap l ) = 0 This is the required equation of the cylinder. Example 4. ind the equation of a right circular cylinder described on the circle through the three points (, 0, 0), (0,, 0), (0, 0, ) are guiding circle. Sol. Let the given three points A (, 0, 0), B (0,, 0), C (0, 0, ). The equation of the sphere OABC is x + y + z x y z = 0 and the equation of the plan ABC is x + y + z =
86 ENGNEERNG MATEMATCS Therefore, the equation of the circle ABC is x + y + z x y z = 0 and x + y + z =...() Since, the cylinder is a right circular cylinder, then the axis of the given cylinder is perpendicular to the plane x + y + z. So direction ratio of the axis are (,, ). The generator through (x, y, z ) and parallel to the axis has equation x x y y z z = = = r Any point on this line (r + x, r + y, r + z )lies on the circle (), if r + x, r + y, r + z = or 3r = (x + y + z )...() and (r + x ) + (r + y ) + (r + z ) (r + x + r + y + r + z ) = 0...(3) Multiply by 3 in equation (3) and using (), we get [ (x + y + z )] + (x + y + z )[ (x + y + z )] + 3[ x + y + z ] = 0 or 3 (x + y + z ] (x + y + z ) + = 0 or x + y + z x y + y z x z = The locus of P (x, y, z ) is or x + y + z xy yz zx =. This is required equation of the right circular cylinder. Example 5. ind the tangent plane to the cylinder 3x + 8xy + 5y + x + 7y + 6 = 0 at the point (,, ). Sol. The tangent plane at the point (x, y, z ) to the cylinder ax + hxy + by + gx + fy + c = 0...() is axx + h (xy + x y) + byy + g (x + x ) + f (y + y ) + c = 0 The tangent plane the given cylinder at (,, ) is 3x () + 4[x ( ) + (y)] + 5y ( ) + (x + ) + 7 (y ) + 6 = 0 or x 5y + 6 = 0 This is the required equation of the tangent plane. Example 6. ind the equation of the sphere enveloping cylinder of the sphere x + y + z x + 4y = 0 having its generation parallel to the line x = y = z. Sol. The equation of the given sphere is x + y + z x + 4y = 0...() the generators of the enveloping cylinder are parallel to the line x = y = z...()
CYLNDER 87 let us consider a point P (x, y, z ) on the given enveloping cylinder. The equation of the generator through the point P (x, y, z ) and parallel to the line x = y = z is x x y = y = z z = r...(3) the point of intersection of the line (3) and the given () are given by (r + x ) + (r + y ) + (r + z ) (r + x ) + 4(r + y ) = 0 or 3r + (x + y + z + )r + (x + y + z x + 4y ) = 0...(4) or enveloping cylinder, the equation (4) must have equal roots. This requires (x + y + z + ) = 3(x + y + z x + 4y ). or x + y + z y z x y 4x + 5y z = 0 The locus of P (x, y, z ) is x + y + z yz xy 4x + 5y z = 0 This is the required equation of the enveloping cylinder. Example 7. ind the equation of the right circular cylinder whose axis is x = z, y = 0 and which passes through the point (3, 0, 0). Sol. The given equation of the axis of the cylinder is x y 0 z = = 0...() 0 We know r = the length of the perpendicular for a point (3, 0, 0) on the cylinder to the axis () = [( 0 0. 0) + {. 03 ( )} + { 0.( 3). 0} ] = () + 0 + Let us consider a point P (x, y, z ) on the cylinder. The length of the perpendicular from the point P to the given axis () is equal to the radius of the cylinder. i.e., {. y 0. z} + {. z. (x )} + {0. (x ). y} = y + (z x + ) + y = x + y + z zx 4x + 4z + 3 = 0 This is the required equation of the right circular cylinder. G K J ( + 0 + ) EXERCSE 3.. ind the equation of the cylinder whose generators are parallel to the line x/ = y/ = z/3 and whose guiding curve is the ellipse x + y =, z = 0.
88 ENGNEERNG MATEMATCS. ind the equation of the cylinder whose generators are parallel to the line x/ = y/ = z/3 and whose guiding curve is the ellipse x + y =, z = 3. 3. ind the equation of the cylinder whose generatoring lines have the direction cosines l, m, n and which passes through the fixed circle is the ellipse x + z = in the ZOX plane. 4. ind the equation of the cylinder whose generators are parallel to the line x/ = y/ = z/3 and passes through the curve x + y = 6 and z = 0. 5. ind the equation of the cylinder whose generators are parallel to the line x/4 = y/ = z/3 and which intersects the ellipse 4x + y =, z = 0. 6. ind the equation of the surface generated by a straight line which is parallel to the line y = mx, z = nx and intersects the ellipse x /a + y /b =, z = 0. 7. ind the equation of the cylinder with generators parallel to z-axis and passes through the curve ax + by = cz, lx + my + nz = p. 8. ind the equation of the cylinder with generators parallel to x-axis and passing through the curve ax + by = cz, lx + my + nz = p. 9. ind the equation of the right circular cylinder of radius whose axis is the line (x )/ = y/3 = (z 3)/. 0. ind the equation of the right circular cylinder whose axis is (x )/ = (y ) = z/3 and which passes through (0, 0, ).. ind the equation of the right circular cylinder of radius 4 whose axis is the line x =y = z.. ind the equation of the right circular cylinder of radius 3 whose axis is the line (x )/ = (y 3)/ = (z 5) / 7. 3. ind the equation of the right circular cylinder whose guiding circle is x + y + z = 9, x y + z = 3. 4. Show that the coordinate of the foot of perpendicular from a point P (x, y, z ) on the line x = y = z are 3 (x + y z ), 3 (x + y z ), 3 (x + y z ). 5. ind the equation of the right circular cylinder whose guiding circle passes through the points (a, 0, 0), (0, b, 0), (0, 0, c). 6. ind the equation of the right circular cylinder with generators parallel to z-axis and intersect the surfaces ax + by + cz =, lx + my + nz = p. 7. ind the equation of the right circular cylinder of radius whose axis is the line (x )/ = (y ) = (z 3)/. 8. ind the equation of the right circular cylinder whose one section is the circle x + y + z x y z = 0, x + y + z = is x + y + z yz zx xy =. 9. ind the equation of the right circular cylinder of radius whose axis passes through (,, 3) and has direction cosines proportional to, 3, 6. 0. ind the equation of the cylinder whose generating line are parallel to the line x/l = y/m = z/n and which touches the sphere x + y + z = a.. ind the enveloping cylinder of the sphere x + y + z = having its generators parallel to the line x/ = y/ = z/3.. Show that the enveloping cylinder of the conicoid ax + by + cz = with generators perpendicular to the z-axis meets the plane z = 0 in parabolas.
CYLNDER 89 3. ind the equation of a right circular cylinder which envelopes a sphere of centre (a, b, c) and radius r, and has its generator parallel to a line with direction ratio l, m, n. 4. ind the equation of the right circular cylinder of radius 5 and having for its axis the line x/ = y/3 = z/6. 5. ind the equation of a cylinder whose generator touches the sphere x + y + z + ux + vy + wz + d = 0 whose generators are parallel to the line x/l = y/m = z/n. 6. Prove that the enveloping cylinder of ellipsoid x y z = = =,whose generator are parallel to line a b c x y z = = meet the plane z = 0 in circles. 0 ± a b c ANSWERS. 9 (x + y + z ) 6xz + 4yz = 9.. 3x + 6y + 3z + 8yz zx + 6x 4y 8z + 4 = 0. 3. (mx ly) + (mz ny) = m. 4. 9x + 9y + 5z 6zx yz 44 = 0. 5. 4 (3x 4z) + (3y + z) = 9. 6. b (nx z) + a (ny mz) = a b n. 7. anx + bny + c(lx + my) pc = 0. 8. (am + bl ) y + an z + amnyz apmy (apn + cl )z = 0. 9. 0x + 5y + 3z xy 6yz 4zx 8x + 30y 74z + 59 = 0. 0. 0x + 3y + 5z 4xy 6yz zx 36x 8y + 30z 35 = 0.. 5x + 8y + 5z 4xy + 4yz + 8zx 44 = 0.. 5x + 5y + 8z 8xy + 4yz + 4zx 6x 4y 96z + 5 = 0. 3. 8x + 5y + 5z + 4xy + 8yz 4zx 7 = 0. x y z x y z 5. + + a b c K (x + y + z ax by cz) = + + a b c + + K a b c 6. (an + cl ) x + (bn + cm ) y + clmny cplx cpmy + cp n = 0. 7. 5x + 8y + 5z 4xy 4yz 8zx + x 6y 4z 0 = 0. 9. 45x + 40y + 3z 36yz 4zx + xy 4x 80y 6z + 94 = 0. 0. (lx + my + nz) = (l + m + n ) (x + y + z a ).. (x + y + 3z) = 4 (x + y + z ).. ab (mx ly) = (al + bm ), z = 0. 3. {l (x a) + m (y b) + n (z c)} = (l + m + n ){(x a) + (y b) + (z c) r }. 4. 45x + 40y + 3z xy 36yz 4zx 5 = 0. 5. {l (x + u) + m (y + v) + n (z + w)} = (l + m + n ) ( x + y + z + ux + vy + wz + d). 6. x + y = a, z = 0.