Maths-
General
Easy
Question
If a sphere of constant radius k passes through the origin and meets the axis in A, B, C then the centroid of the triangle ABC lies on
- 9 (x2 + y2 + z2)= k2
- 9 (x2 + y2 + z2)= 4k2
- x2 + y2 + z2 = k2
- x2 + y2 + z2 = 4k2
The correct answer is: 9 (x2 + y2 + z2)= 4k2
In this question we should find the locus of the centroid of ABC
![L e t space t h e space e q u a t i o n space o f space t h e space s p h e r e space b e space space left parenthesis x minus alpha right parenthesis squared space space plus left parenthesis y minus beta right parenthesis squared space plus left parenthesis z minus gamma right parenthesis squared space equals k to the power of 2 end exponent
space S i n c e space i t space p a s s e s space t h r o u g h space t h e space o r i g i n space space
therefore alpha squared space space plus beta squared space plus gamma squared equals space k squared space space space a n d space m e e t s space t h e space c o o r d i n a t e space a x i s space a t space t h e space p o i n t space A left parenthesis 2 alpha comma 0 comma 0 right parenthesis comma beta left parenthesis 0 comma 2 beta comma 0 right parenthesis comma c left parenthesis 0 comma 0 comma 2 gamma right parenthesis space
T h e space c e n t r o i d space o f space t h e space t r i a n g l e space A B C space i s space space left parenthesis space space 2 space 2 alpha space space space comma space space 3 space 2 beta space space space comma space space 3 space 2 gamma space space space right parenthesis equals left parenthesis u comma v comma w right parenthesis space
T h e n space u squared space space plus v squared space space plus w squared space space equals left parenthesis space 9 space 4 space right parenthesis left parenthesis alpha squared space space plus beta squared space plus gamma squared space space right parenthesis equals left parenthesis space 9 space 4 space space right parenthesis equals k squared space space space space
T h u s space left parenthesis u comma v comma w right parenthesis space l i e s space o n space 9 left parenthesis x to the power of 2 to the power of blank end exponent space space plus y squared space space plus z squared space space right parenthesis equals 4 k 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)
Hence option 2 is correct
Related Questions to study
Maths-
The equation of the sphere circumscribing the tetrahedron whose faces are x = 0, y = 0, z = 0 and
is equal to
The equation of the sphere circumscribing the tetrahedron whose faces are x = 0, y = 0, z = 0 and
is equal to
Maths-General
Maths-
The lines
and
are coplanar if l is
The lines
and
are coplanar if l is
Maths-General
Maths-
The line
intersects the curve xy = c2, z = 0 if c =
The line
intersects the curve xy = c2, z = 0 if c =
Maths-General
Maths-
Maths-General
Maths-
Equation of the plane through the points (2, 1, -1) and (-1, 3, 2) and perpendicular to the plane x – 2y + 4z = 0 is given by
Equation of the plane through the points (2, 1, -1) and (-1, 3, 2) and perpendicular to the plane x – 2y + 4z = 0 is given by
Maths-General
Maths-
Maths-General
Maths-
Maths-General
Maths-
The four points (0, 4, 3) , (-1, -5, -3) , (-2, -2, 1) and (1, 1, -1) lie in the plane
The four points (0, 4, 3) , (-1, -5, -3) , (-2, -2, 1) and (1, 1, -1) lie in the plane
Maths-General
Maths-
The equation of a plane and condition of two planes being parallel or perpendicular.
A) Every equation of first degree in x, y, z, i. e., Ax + By + Cz + D = 0 represents a plane. The coefficients of x, y, z are the direction ratios of the normal to the plane.
B) Angle between two planes is equal to the angle between the normals to the planes.
\ cos q = ![fraction numerator A subscript 1 end subscript A subscript 2 end subscript plus B subscript 1 end subscript B subscript 2 end subscript plus C subscript 1 end subscript C subscript 2 end subscript over denominator square root of not stretchy sum A subscript 1 end subscript superscript 2 end superscript end root. square root of not stretchy sum A subscript 2 end subscript superscript 2 end superscript end root end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ4AAAA6CAYAAABBCOCDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAABRhJREFUeNrtXV+ITUEYnzZJkijbkmSTtEnyoETkZZO0D5tIG5KUtHnwsIVaUZKSEErStm2SkhAbD5IkKW3IA0WSJKQ2+bv5d33f+m7G6e45Z+Z8586cM9+vfrWde8/vnt/MnJlv5pz5VikzbFO84NazxRDwJ/Al8DnwMnCKQx0uzAAeAD4EfqXr2guc6IleKqwBfgeO8lTPFjOpIHVgYfY60uFCF7AfuATYQMem0c3e64FeKmCLfgS8Dmz3UE9HxfD7+PtnIsdW1jhWLx0OT9uBpxjLlFsvNU4CO4AbgMc91MtSSbupYHX0AZc70snqaRYN81wjCbdeaiykLhbRRBfhk17WhneOeiscPuYBeyxjTy6drJ4OMsfO3HqpgK18gMbyKu4DWzzR42h4T+kc5OcMPRSXTlZPjyne5AK3XurhoytybB+w0xM9pVV2HEdCAzUSxGhgK/AuDS8m4NLJ6qmBZtZc4NZLhRbqjaJYSssErvU4eocFwBuRY2tpyUA50MnqCRv9R8ay5NZLhdsxd1utZRC8u3fVWFKw0dOP3zHs6k0aXgfFYjrW15jBJXlLo1MvT9jzjjWcPMR5M9HL4nEYm4FHYj4/C1wROXaazqsw6VW7+s4RekqOSjoK3Bg51kMNKa03E516eOqpEc7EIcmbqZ6tx+GVdlxjGxfznU0xDanCrKcM4wyTSrqgTQIagavpzh9tqG2qk6en6cBB4E7174nCGGAbNaJWw9+w1TP1OLws0Jbwnak0i0tjIKsexoC3cgonBrXhYYh63qkWDcBUJ09Pioa4PorP8Jo+AM+PMKqkadw2enl7zLyckdT7YqwwQ/mBSgk9labeuAxg0Nuv3D5k5/bmo6fS1BtXr3AVOL5EleOrp9LUG4cBvPiWklWOr54KX28mTwzqqeWLN988hVBvAoFAIBAIBAKPgkyhkJsCgUAgEAgEAgErxgUcIIfs3Tlw+9+zQG+6kL07x1blLrWDeA8YuAmmS7wL6o0nwPniXVDvicUnFb/5R7wL2LEY+CDmc9ezrsx7Q8W7n+hS8SmvfJnuW+0NFe/+AjcQb4n5fJNWkIcs7lhuDIn3cuBFiuB6mfq3hxPzqoxxVPjce0ND9u48uP6h0iX5mw18TQWKqc0ama8lKecb997QkL07B2YUuGnw/SYqeKyAV1QhHEjKv5zH3tCQvTvHDuAxw3PG0pBTTZ2wLOM1JOVfzmtvaMjeneMicJ3luYeoAn5SEG6LpPzLee0NDdm7c+CQMcfy3A5txtdjqZEm/3JeyxQhe3eKycD3yi57+GxtpjdgMNPTUY/8y+K9BBMLfTZYTWw9GCk8E+SRf1m8e9i7RbEHuN9C65LW7dsG1/XKvxy6d2fAZ3u9VFDRjOd4x68y1NuuFfzuDNdlms9ZvBcM3VT4X9TfvMA6MMZpNtBaqhVQf4Zrss2/LN4LiCvq/9e7sTd4Y3B3YRrXd1TwOPuy/e+AtvmXkzKhR2eBZfFe+LdTcHHyhzYDw2HmmsEM7C4VAP5ryrkGv8uVfzkpE3pcwyu6d0Sh3075pXXjh2koSoOTWoWuNfxN7nWnSsDeEYV8O+UZDTvV4LrdoAB92X5XCdh7Yd9OOQF8S39/A04KqOEV3Xuh307B+OQ3cJH6+7ioiKgE6L0Ub6d8psD6bGANr6jeS/N2yj2qvO4AG14RvZfm7ZRuKvzWAja4rPFTEb2X5u2UCXTxE1R4CNm7Fzgm3gUu0CzeBVH8AZiqyOzVP1q3AAAB7nRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZnJhYz48bXJvdz48bXN1Yj48bWk+QTwvbWk+PG1uPjE8L21uPjwvbXN1Yj48bXN1Yj48bWk+QTwvbWk+PG1uPjI8L21uPjwvbXN1Yj48bW8+KzwvbW8+PG1zdWI+PG1pPkI8L21pPjxtbj4xPC9tbj48L21zdWI+PG1zdWI+PG1pPkI8L21pPjxtbj4yPC9tbj48L21zdWI+PG1vPis8L21vPjxtc3ViPjxtaT5DPC9taT48bW4+MTwvbW4+PC9tc3ViPjxtc3ViPjxtaT5DPC9taT48bW4+MjwvbW4+PC9tc3ViPjwvbXJvdz48bXJvdz48bXNxcnQ+PG1vPiYjeDIyMTE7PC9tbz48bXN1YnN1cD48bWk+QTwvbWk+PG1uPjE8L21uPjxtbj4yPC9tbj48L21zdWJzdXA+PC9tc3FydD48bW8+LjwvbW8+PG1zcXJ0Pjxtbz4mI3gyMjExOzwvbW8+PG1zdWJzdXA+PG1pPkE8L21pPjxtbj4yPC9tbj48bW4+MjwvbW4+PC9tc3Vic3VwPjwvbXNxcnQ+PC9tcm93PjwvbWZyYWM+PC9tYXRoPlIwAMwAAAAASUVORK5CYII=)
Planes are perpendicular if å A1 A2 = 0 and parallel if ![fraction numerator A subscript 1 end subscript over denominator A subscript 2 end subscript end fraction equals fraction numerator B subscript 1 end subscript over denominator B subscript 2 end subscript end fraction equals fraction numerator C subscript 1 end subscript over denominator C subscript 2 end subscript end fraction.](data:image/png;base64,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)
C) Planes parallel to co – ordinate planes are x = l, y = l or z = l.
Planes perpendicular to co – ordinate planes x = 0, y = 0, z = 0 are
by + cz + d = 0 (x missing), ax + cz + d = 0
(y missing), ax + by + d = 0 (z missing) Find the equation of the plane through the intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is
The equation of a plane and condition of two planes being parallel or perpendicular.
A) Every equation of first degree in x, y, z, i. e., Ax + By + Cz + D = 0 represents a plane. The coefficients of x, y, z are the direction ratios of the normal to the plane.
B) Angle between two planes is equal to the angle between the normals to the planes.
\ cos q = ![fraction numerator A subscript 1 end subscript A subscript 2 end subscript plus B subscript 1 end subscript B subscript 2 end subscript plus C subscript 1 end subscript C subscript 2 end subscript over denominator square root of not stretchy sum A subscript 1 end subscript superscript 2 end superscript end root. square root of not stretchy sum A subscript 2 end subscript superscript 2 end superscript end root end fraction](data:image/png;base64,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)
Planes are perpendicular if å A1 A2 = 0 and parallel if ![fraction numerator A subscript 1 end subscript over denominator A subscript 2 end subscript end fraction equals fraction numerator B subscript 1 end subscript over denominator B subscript 2 end subscript end fraction equals fraction numerator C subscript 1 end subscript over denominator C subscript 2 end subscript end fraction.](data:image/png;base64,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)
C) Planes parallel to co – ordinate planes are x = l, y = l or z = l.
Planes perpendicular to co – ordinate planes x = 0, y = 0, z = 0 are
by + cz + d = 0 (x missing), ax + cz + d = 0
(y missing), ax + by + d = 0 (z missing) Find the equation of the plane through the intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is
Maths-General
Maths-
Maths-General
Maths-
The direction cosines of the line joining the points (4, 3, - 5) and (-2, 1, -8) are
For such questions, we should know formula to find direction cosines.
The direction cosines of the line joining the points (4, 3, - 5) and (-2, 1, -8) are
Maths-General
For such questions, we should know formula to find direction cosines.
Maths-
The locus of x2 + y2 + z2 = 0 is
For such questions, we have to be careful about points satisfying the equation.
The locus of x2 + y2 + z2 = 0 is
Maths-General
For such questions, we have to be careful about points satisfying the equation.
Maths-
Maths-General
Maths-
Maths-General
Maths-
Let f (x) be a quadratic expression which is positive for all real x. If g(x) = f (x) + f ¢ (x) + f ¢¢ (x), then for any real x
Let f (x) be a quadratic expression which is positive for all real x. If g(x) = f (x) + f ¢ (x) + f ¢¢ (x), then for any real x
Maths-General