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Three Dimensional Geometry (part - 2) , The Planes | study material for IIT JEE | concept booster, chapter highlights/

THREE DIMENSIONAL GEOMETRY (PART - 2), THE PLANES










THE PLANESPlane is the surface such that if any two points are taken on it, then the line joining the two points lies on it.
General equation of plane is given as -  ax + by + cz + d = 0 , where a ,b and c is not equal to zero.
           POINTS TO BE REMEMBERED 1). a, b and c are the directions ratios of the normal to the plane ax + by + cz + d = 0.

2). Equation of the yz - plane is x = 0

3). Equation of the zx - plane is y = 0

 4). Equation of the xy - plane is z = 0

5). Equation of the any plane parallel to the xy - plane is z = c. Similarly for Planes parallel to yz and zx is x = c and y = c
EQUATION OF THE PLANE IN NORMAL FORMVECTOR FORMIf n̂ be a unit vector normal to a given plane and d be the length of the perpendicular from the origin to the plane, then the equation of the plane is given by -                           r.n̂ = d CARTESIAN FORMIf l, m, n are the directions cosines of the normal to the plane and d is the perpendicular distan…

Partnership & Work and wages problems tricks in Hindi | fast track formulae for problem solving.

WORK AND WAGES PROBLEMS TRICKS IN HINDI

Work and wages problems tricks in Hindi | fast track formulae for problem solving.

1). कुल वेतन = कुल दिन × 1 दिन का वेतन

2). यदि X , Y, Z किसी कार्य को d1 , d2 , d3 दिनों में कर सकते हैं, तो उनके वेतनो का अनुपात = 
     d2d3 : d3d1 : d1d2 

3). यदि A और B किसी कार्य को x और y दिनों में कर सकते हैं, तथा उनके वेतनो का अनुपात y : x है, तो A और B के वेतन क्रमशः , A = y [कुल वेतन/(x+y)] ,
             B = x[कुल वेतन/(x+y)]

4). यदि A , B और C किसी कार्य को x,y व z दिनों में कर सकते हैं, तथा उनके वेतनो का अनुपात yz : xz : xy है, तब A , B और C के वेतन क्रमशः -

A = yz [कुल वेतन/(yz : xz : xy)]
B = zx [ कुल वेतन/(yz : xz : xy)]
C =  xy [ कुल वेतन/(yz : xz : xy)]

5). निश्चित व्यक्ति द्वारा निश्चित कार्य करके कमाया गया कुल वेतन = (1 व्यक्ति का 1 दिन का वेतन) × ( व्यक्तियों की संख्या) × (दिनों की संख्या).
 
तो आवश्यक व्यक्तियों की संख्या =
 कुल वेतन/(1 व्यक्ति का 1 दिन का वेतन) × (दिनों की संख्या).

6). A किसी कार्य को x दिन में कर सकता है, तथा B की सहायता से A किसी कार्य को y दिन में कर सकता है, तब A का हिस्सा तथा B का हिस्सा , यदि वे ₹ a प्राप्त करते है। = 

A का हिस्सा = ₹ ay/x
B का हिस्सा = ₹ a (x - y)/x

7). यदि X, Y, Z किसी दिए गए कार्य को ₹ R में करने का बेड़ा उठाते हैं, यदि X और Y दोनों मिलकर सम्पूर्ण कार्य का केवल m/n ही हिस्सा करते पाते है, तथा शेष अकेला Z करता है, तब Z का हिस्सा = ₹ R (n - m)/n

PARTNERSHIP PROBLEMS TRICKS IN HINDI

Partnership problems tricks in Hindi | fast track formulae for problem solving.

1). साझेदारी द्वारा निवेश की गई धनराशि का अनुपात = लाभ के भागो का अनुपात

2). यदि कुछ व्यक्ति विभिन्न धनराशियों P1, P2, P3 ... को विभिन्न समयो t1, t2, t3 ... के लिए निवेश करके साझेदारी में व्यापार करते है, तो उनके लाभो का अनुपात = 
   P1t1 : P2t2 : P3t3 : ...

3). यदि साझेदारी के लाभों का अनुपात x : y : z है, तो उनके द्वारा लगाई गई , धनराशियों के समयों का अनुपात = 
  x/p1 : y/p2 : z/p3 : ... तथा धनराशियों का अनुपात = 
   x/t1 : y/t2 : z/t3 : ...

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Three Dimensional Geometry (part - 2) , The Planes | study material for IIT JEE | concept booster, chapter highlights/

THREE DIMENSIONAL GEOMETRY (PART - 2), THE PLANES










THE PLANESPlane is the surface such that if any two points are taken on it, then the line joining the two points lies on it.
General equation of plane is given as -  ax + by + cz + d = 0 , where a ,b and c is not equal to zero.
           POINTS TO BE REMEMBERED 1). a, b and c are the directions ratios of the normal to the plane ax + by + cz + d = 0.

2). Equation of the yz - plane is x = 0

3). Equation of the zx - plane is y = 0

 4). Equation of the xy - plane is z = 0

5). Equation of the any plane parallel to the xy - plane is z = c. Similarly for Planes parallel to yz and zx is x = c and y = c
EQUATION OF THE PLANE IN NORMAL FORMVECTOR FORMIf n̂ be a unit vector normal to a given plane and d be the length of the perpendicular from the origin to the plane, then the equation of the plane is given by -                           r.n̂ = d CARTESIAN FORMIf l, m, n are the directions cosines of the normal to the plane and d is the perpendicular distan…

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