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Maths-

General

Easy

Question

The equation of plane passing through a point $A left parenthesis 2 comma negative 13 right parenthesis$ and parallel to the vectors $a equals left parenthesis 30 comma negative 1 right parenthesis$ and $b equals left parenthesis negative 32 comma 2 right parenthesis$ is

$2 x - 3 y + 6 z - 25 = 0$
$2 x - 3 y + 6 z + 25 = 0$
$3 x - 2 y + 6 z - 25 = 0$
$3 x - 2 y + 6 z + 25 = 0$

The correct answer is: $2 x minus 3 y plus 6 z minus 25 equals 0$

As plane is parallel to a given vector Þ Normal of plane must perpendicular to the given vectors. Given point to which plane passes through is (2, –1,3).
Let A, B,C are direction ratios of its normal.
\ Equation of plane is, $A left parenthesis x minus 2 right parenthesis plus B left parenthesis y plus 1 right parenthesis plus C left parenthesis z minus 3 right parenthesis equals 0$
…..(i)

Now normal to plane $A i plus B j plus C k$ is perpendicular to the given vectors $a equals 3 i plus 0 j minus k$ and $b equals negative 3 i plus 2 j plus 2 k$
\ $3 A plus 0 B minus C equals 0$ …..(i)
$negative 3 A plus 2 B plus 2 C equals 0$ .....(ii)
Solving (i) and (ii) we get, $fraction numerator A over denominator 2 end fraction equals fraction numerator B over denominator negative 3 end fraction equals fraction numerator C over denominator 6 end fraction$
\Equation of plane be $2 left parenthesis x minus 2 right parenthesis minus 3 left parenthesis y plus 1 right parenthesis plus 6 left parenthesis z minus 3 right parenthesis equals 0$
i.e., $2 x minus 3 y plus 6 z minus 25 equals 0$ .

Related Questions to study

The equation of the plane passing through the points $left parenthesis negative 1 comma negative 2 , 0 right parenthesis comma left parenthesis 2 , 3 comma 5 right parenthesis$ and parallel to the line
$r equals negative 3 j plus k plus lambda left parenthesis 2 i plus 5 j minus k right parenthesis$ is

The equation of the plane passing through the points $left parenthesis negative 1 comma negative 2 , 0 right parenthesis comma left parenthesis 2 , 3 comma 5 right parenthesis$ and parallel to the line
$r equals negative 3 j plus k plus lambda left parenthesis 2 i plus 5 j minus k right parenthesis$ is

The distance from the point $negative i plus 2 j plus 6 k$ to the straight line through the point (2, 3, –4) and parallel to the vector $6 i plus 3 j minus 4 k$ is

The distance from the point $negative i plus 2 j plus 6 k$ to the straight line through the point (2, 3, –4) and parallel to the vector $6 i plus 3 j minus 4 k$ is

The length of the perpendicular from the origin to the plane passing through three non-collinear points $a comma b comma c$ is

The length of the perpendicular from the origin to the plane passing through three non-collinear points $a comma b comma c$ is

parallel

A vector $n$ of magnitude 8 units is inclined to x-axis at $4 5 to the power of o end exponent$ , y-axis at $6 0 to the power of o end exponent$ and an acute angle with z-axis. If a plane passes through a point $left parenthesis square root of 2 comma negative 1 , 1 right parenthesis$ and is normal to $n$ , then its equation in vector form is

A vector $n$ of magnitude 8 units is inclined to x-axis at $4 5 to the power of o end exponent$ , y-axis at $6 0 to the power of o end exponent$ and an acute angle with z-axis. If a plane passes through a point $left parenthesis square root of 2 comma negative 1 , 1 right parenthesis$ and is normal to $n$ , then its equation in vector form is

$fraction numerator text d end text over denominator text dx end text end fraction open square brackets s i n open parentheses l o g open parentheses s i n e to the power of x to the power of 2 end exponent end exponent close parentheses close parentheses close square brackets$ =

$fraction numerator text d end text over denominator text dx end text end fraction open square brackets s i n open parentheses l o g open parentheses s i n e to the power of x to the power of 2 end exponent end exponent close parentheses close parentheses close square brackets$ =

If $square root of T a n y end root$ = $e to the power of c o s 2 x end exponent. s i n x comma blank t h e n blank fraction numerator text dy end text over denominator text dx end text end fraction$ =

If $square root of T a n y end root$ = $e to the power of c o s 2 x end exponent. s i n x comma blank t h e n blank fraction numerator text dy end text over denominator text dx end text end fraction$ =

parallel

The line $chi plus gamma equals 2$ is tangent to tthe curve $chi to the power of 2 end exponent equals 3 minus 2 gamma$ at its point

The line $chi plus gamma equals 2$ is tangent to tthe curve $chi to the power of 2 end exponent equals 3 minus 2 gamma$ at its point

In general calcium carbonate is not soluble in water but by the addition of carbon dioxide to it, it becomes soluble. This is because:

In general calcium carbonate is not soluble in water but by the addition of carbon dioxide to it, it becomes soluble. This is because:

The correct stability order for the following species is:
I)
II)
III)
IV)

The correct stability order for the following species is:
I)
II)
III)
IV)

Chemistry-General

parallel

The point (4,3) is translated to the point (3,1) and then axes rotated through 30º about the origin then the new position of point is

The point (4,3) is translated to the point (3,1) and then axes rotated through 30º about the origin then the new position of point is

The arrangement of the following in ascending order of angle of rotation to eliminate xy term in the following equations.
i) 2 $text x end text to the power of 2 end exponent$ +5xy+2 $text y end text to the power of 2 end exponent$ +5x+7y+1=0
ii) 2 $text x end text to the power of 2 end exponent$ + $square root of text 3 end text end root$ xy+3 $text y end text to the power of 2 end exponent$ = 9
iii) 9 $x to the power of 2 end exponent$ +2 $square root of text 3 end text end root$ xy+7 $text y end text to the power of 2 end exponent$ = 10

The arrangement of the following in ascending order of angle of rotation to eliminate xy term in the following equations.
i) 2 $text x end text to the power of 2 end exponent$ +5xy+2 $text y end text to the power of 2 end exponent$ +5x+7y+1=0
ii) 2 $text x end text to the power of 2 end exponent$ + $square root of text 3 end text end root$ xy+3 $text y end text to the power of 2 end exponent$ = 9
iii) 9 $x to the power of 2 end exponent$ +2 $square root of text 3 end text end root$ xy+7 $text y end text to the power of 2 end exponent$ = 10

If the distance between the two given points is 2 units and the points are transferred by shifting the origin to (2,2) then the distance between the points in their new position is

If the distance between the two given points is 2 units and the points are transferred by shifting the origin to (2,2) then the distance between the points in their new position is

parallel

The transformed equation of $7 x to the power of 2 end exponent plus 2 square root of 3 x y plus 9 y to the power of 2 end exponent equals 8$ when the axes are rotated through an angle $fraction numerator text π end text over denominator text 3 end text end fraction$ is

The transformed equation of $7 x to the power of 2 end exponent plus 2 square root of 3 x y plus 9 y to the power of 2 end exponent equals 8$ when the axes are rotated through an angle $fraction numerator text π end text over denominator text 3 end text end fraction$ is

The transformed equation of $x to the power of 2 end exponent minus y to the power of 2 end exponent plus 2 x plus 4 y equals 0$ when the origin is shifted to the point (-1,2) is

The transformed equation of $x to the power of 2 end exponent minus y to the power of 2 end exponent plus 2 x plus 4 y equals 0$ when the origin is shifted to the point (-1,2) is

The point to which the origin should be shifted in order to eliminate x and y terms in the equation $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent minus 8 x plus 36 y plus 4$ = 0

The point to which the origin should be shifted in order to eliminate x and y terms in the equation $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent minus 8 x plus 36 y plus 4$ = 0

parallel

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