ANALYTICAL GEOMETRY (3D) AND INTEGRAL CALCULUS

ANALYTICAL GEOMETRY (3D) AND INTEGRAL CALCULUSUNIT IStandard equation of a plane – intercept form-normal form-plane passing through givenpoints – angle between planes –plane through the line of intersection of two planes- Equation of thestraight line – Shortest distance between two skew lines- Equation of the line of shortest distanceUNIT IISphere – Standard equation –Length of a tangent from any point-Sphere passing through agiven circle – intersection of two spheres – Tangent plane.UNIT IIIIntegration by parts – definite integrals & reduction formulaUNIT IVDouble integrals – changing the order of Integration – Triple Integrals.UNIT VBeta & Gamma functions and the relation between them –Integration using Beta & GammafunctionsTEXT BOOK(S)[1] T.K.Manickavasagam Pillai & others, Analytical Geometry, S.V Publications -1985 RevisedEdition.[2] T.K.Manickavasagam Pillai & others, Integral Calculus, SV Publications.UNIT – I - Chapter 2 Sections 13 to 21 & Chapter 3 Sections 24 to 31 of [1]UNIT – II - Chapter 4 Sections 35 to 42 of [1]UNIT – III - Chapter 1 Sections 11 , 12 & 13 of [2]UNIT – IV - Chapter 5 Sections 2.1 , 2.2 & Section 4 of [2]UNIT – V - Chapter 7 Sections 2.1 to 5 of [2]REFERNECE(S)[1] Duraipandian and Chatterjee, Analytical Geometry[2] Shanti Narayan, Differential & Integral Calculus.1

PART – A1. Write down the formula to find the angle between PlanesandThe angle between two planes is2. Write down the Condition for the lineto be Parallel to the PlaneCondition for the line to be parallel to the plane is3. Formula for the Length of the tangent from the Pointtothe SphereThe length of the tangent from a point to the sphere is4. Writethe formula for equation of Tangent plane to SphereThe equation of tangent plane to sphere is5. Write down the intercept form of the equation of a planeThe intercept form of the plane is6. Write down the Condition for two given straight lines to be coplanar.2

The condition for two linesto be coplanar is7. Find the centre and radius of the SphereThe centre is c( u, v, w)The radius8. Write down the formula for intersection of two spheres:The intersection of two sphere is9. Find the equation of the plane through Pt(2, -4, 5) and parallel to the plane4x 2 y 7z 60Equation of a plane parallel toGiven 1 passes thro10. Find the point where the line3

meets the plane xyz5Let the given line be11. Find the Centre and Radius of the SphereGivenRadius12. Find the equation of the Sphere thro the circle2x 3 y 4z5 and point (1,2,3)4

Given :sx2yz290 ; 2x 3 y 4z 501 passes thro (1,2,3);13. Write the equation of the plane making interceptsGiven a 2,b 3,c 4Equation of plane in intercept form is14. Find the length of the Tangent from the pointto the SphereGIVEN :The Length of The Tangent is5

15. Write down the Equation of the Planeand Parallel to the lineThe Equation of the Plane Containing the Line16. What are the Intercepts of17. Find the Length of Perpendicular drawn from (1, 2, 3) on6

Given:18. Find the Length of the Tangentto the SphereGiven : 7

19. State the Condition for the Straight Lineto lie on the planeCondition for line to lie on the plane:The line should be parallel to the plane andany point on the line should pass through the plane.20. Find the tangent plane at (-1, 4, -2) onThe equation of the tangent plane isThe equation of tangent plane at (-1, 4,-2) onis21. Find the normal form ofGiven :8

22. Evaluate:Solution:9

23. Evaluate:Solution24. Write down the recurrence formula for ГnSolution:Recurrence formula for Гn10

25. . Evaluate:Solution:26. EvaluateSolution:11

27. Find the value of ГnSolution:28. Find the value of ГSolution:.29. Evaluate:12

30.31. Write down the formula for relation between beta and gamma function13

11. Prove thatГFrom 1 and 214

.12.Solution:By the property of definite integrals13. Find the value of Г0‘n’ is a positive integer.Г015

14. State the recurrence formula for gamma function.PART – B16

1. Find the shortest distanceSolution: Let Any point on line isAny point on line isThe direction of line joining the point G & HThe direction’s of Line (3,-1,1)We’ve Also the direction of line is (-3,2,4)We’ve17

Solving & X 7 77 X 11 -77-270Subin Shortest distance between G & H is18

2. Find the centre and radius of the circleSolution:Given the equation of the sphere iscraThe direction of CQ is (x-0,y-1,z-2)The direction ofThe line CQ isto the given plane and hence parallel to the normal to the planeThe direction of proportionalThe point Q is (K, 2K 1,2K 2)The point is lies on the plane19

The point Q is (1,3,4)Distance between two pointsRadius of the circle3. Derive the intercept form of a planeStatement :find the equation of a plane making intercepts a,b.c on the axis ox, oy, oz respectivelyProof:let the given plane meet the co-ordinate axis ox, oy, oz, at A, B, C respectively20

Hence the co-ordinates of the points A,B,C are respectively (a,0,0) , (0,b,0), (0,0,c)Subs the values inThis is known as intercept form of the equation of a plane.4. Find the equation of the plane through the lineSolution:Given that Equation of the plane passing through line is 21

Since it is parallel to the line Solving & 5. Obtain the equation of the sphere through the origin and the circle given bySolution:Equation of the sphere passing through the given circle is 22

Since it passes through the origin (0,0,0) 6. Show that the planeSolution:GivenCentre (-u, -v, -w)Centre (1, 2, -1)The perpendicular distance from the centre to the plane23

Perpendicular distance 3 unitsFrom & The 24

7. ProveSolution:Adding 1 & 2 we get25

To evaluateSub 4 in 3 we get26

8. EvaluateSolution:9. Evaluate27

10. Evaluate :Solution:WKT28

Put29

11.Evaluate:Solution:30

12. Evaluate:Solution:Adding 1 & 2 we get31

32

13. IfSolution:Sub n by n-1 inMultiply ‘a’ on both sidesSubtract 2 from 114. IfSolution:33

15. Obtain a reduction formula for34

Solution:35

16. Evaluate:Solution:n 3, m 2 in36

Sub 3 in 2 we’ve17. Show thatSolution:37

38

18. Show thatSolution:WKTConsider LHS19. Prove .That ГSolution:WKT39

WKT40

From 1 & 2.20. Evaluate:ySolution:x 0xy 041

Put42

21. Evaluatetaken over the positive of the circlesolution:But given positive quadrant.43

PART – C1. Show that the linesand 3x – 2y z 5 0 and2x 3y 4z – 4 0 lies on a plane, also. Find the point of intersection and the equationof the plane.Solution:To find the symmetric form of 3x – 2y z 5 0 & 2x 3y 4z – 4 0The equation of the plane are Solving 1 & 2in44

Any point on the given line isStep :2To find direction ratio’s(using the condition of perpendicular )Solving & 45

(a, b, c)( 11, 10,13)Condition for co-planer 0 046

0 0The line are Co-planar Let Any point on the line isAny point on the line isStep: 3To find the point of intersectionComparing x terms47

Comparing y terms Comparing z terms Solving & X348

SubSubin in The lines are intersectingThe point of intersecting is (2, 4, -3)Equation of the plane 049

02. Find the condition that the plane lx my nz 8 may touch the sphereSolution:A plane will touch a sphere if the length of from the centre of the sphere to the plane isequal to the radius of the sphere.The centre of the given sphere isand the radius isNow the length of the perpendicular fromtothe planeLength of the perpendicular radius of the sphere50

Squaring on both sides 3. Find the equation of the tangent plane to the sphereat p(x, y, z) on it.Also S.T radius to the radius through the point.Solution: The given sphere isW.K.T the line joining the centre of a sphere to any point on point it isto thetangent plane at the point.The centre of the given sphere isthe direction ration of the line joiningthe pointThe equation of the plane passing through and having cp as its normal is,51

Butis a point on the sphere And therefore satisfies (ie)(ie)Substituting the value ofin we getWhich is required equations of the tangent plane.4. Find theS D between z axis and straight line given bySolution :The equation of the plane passing through the lineis The plane given by equation 1 is parallel to z axis The direction cosines of z axis (0, 0, 1)52

The normal to the plane isto z axis by the condition ofThus the equation of the plane isThe equation of the plane is Any point on z axis isto the plane The shortest distance length of the5. Obtain the length the tangent fromSolution: Let p be the pointsince (0, 0, 0) lie on z axisfrom (0, 0, 0) to the plane to the sphere& C be the centre of the sphere53

To prove :A tangent from p to the sphere the co-ordinates of the sphere is equal toTP(x1, y1, z1)CCCCC C ( U , V , W )6. EvaluateSolution: 54

From & 55

7. Ifshow thatdeduce the value f(n,n)Solution W.K.T56

2, becomes57

8. EvaluateSolution: ,58

59

9.EvaluateSolution60

61

Hence10. EvaluateSolution:To cover the whole positive octant of the sphereHence the required integral is62

63

11. EvaluateSolution :multiply by abc 64

65

66

a 3b 2 c 2252012. ExpressionSolution67

HereTo find68

12.Proof:69

Similarly ГnMultiply & To evaluate the double integral in . we can use the polar coordinates putthe area of the element dx. dy becomesregion in polar co-ordinates we have to taketo cover thisfrom ‘0’ to From 70

& 13. Evaluate:Solution:-aao-b71

But given that x,y in positive quadrant14.72

x 0y xy 0The region of integration is bounded by lines y x, x 0 ( y axis ) and an infiniteboundaryTake – Strips to x- axis to change the order of integrationThe extremities of the strip lie on x 0 , y x.15.73

yQ(0,2a)x'oxP(a,a)y’x y 2ax varies from 0 to a. hence the region of integration is OPQ .by changing the order of integration, we first integrate x keeping y as constant.Thus the strip is parallel to x axis and x varies .in covering the same region the end of these strips extend to the line y 2a-x tothe curvehence we divide the integration into two parts by the liney a which passes through pin the first region y varies from 0 to a and in the next region from 0 to 2a.74

75

ANALYTICAL GEOMETRY (3D) AND INTEGRAL CALCULUS UNIT I Standard equation of a plane – intercept form-normal form-plane passing through given points – angle between planes –plane through the line of intersection of two planes- Equation of the straight line – Shortest distance between two skew lines- Equation of the line of shortest distance UNIT II Sphere – Standard equation –Length .