Home
Class 12
MATHS
If the plane ax+by=0 is rotated about it...

If the plane `ax+by=0` is rotated about its line of intersection with the plane `z=0` through an angle `alpha` then prove that the equation of the plane in its new position is `ax+by+-(sqrt(a^(2)+b^(2)) tan alpha)z=0`.

Text Solution

Verified by Experts

Given planes are `ax+by=0" "`(i)
and `" "z=0" "`(ii)
Therefore, the equation of any plane passing through the line of intersection of planes (i) and (ii) may be taken as
`" "ax+by+kz=0" "`(iii)
The direction cosines of a normal to the plane (iii) are
`" "(a)/(sqrt(a^(2)+b^(2)+k^(2))),(b)/(sqrt(a^(2)+b^(2)+k^(2))) and (k)/(sqrt(a^(2)+b^(2)+k^(2)))`
The direction cosines of a normal to the plane (i) are
`" "(a)/(sqrt(a^(2)+b^(2))),(b)/(a^(2)+b^(2))) and 0`
Since the angle between the planes (i) and (ii) is `alpha`, we have
`" "cosalpha=(a*a+b*b+k*0)/(sqrt(a^(2)+b^(2)+k^(2))sqrt(a^(2)+b^(2)))=sqrt((a^(2)+b^(2))/(a^(2)+b^(2)+k^(2)))`
or `" "k^(2)cos^(2)alpha=a^(2)(1-cos^(2)alpha)+b^(2)(1-cos^(2)alpha)`
or `" "k^(2)=((a^(2)+b^(2))sin^(2)alpha)/(cos^(2)alpha)or k=pmsqrt(a^(2)+b^(2))tanalpha`
Putting this in (iii), we get the equation of the plane as `ax+bypmzsqrt(a^(2)+b^(2))tanalpha=0`
Promotional Banner

Topper's Solved these Questions

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 3.1|12 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 3.2|15 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise All Questions|294 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE ENGLISH|Exercise Archives (Matrix Match Type)|1 Videos

Similar Questions

Explore conceptually related problems

The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle alpha . Prove that the equation of the plane in its new position is ax + by +- (sqrt(a^2 + b^2) tan alpha)z = 0

The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle alpha . Prove that the equation of the plane in its new position is ax + by +- (sqrt(a^2 + b^2) tan alpha)z = 0

The plane l x+m y=0 is rotated through an angle alpha about its line of intersection with the plane z=0 . Prove that the equation of the plane in its new position is l x+m ypm(sqrt(l^2+m^2)tanalpha)z=0.

The plane ax+by=0 is rotated through an angle alpha about its line of intersection with the plane z=0 . Show that the equation to the plane in new position is ax+bypmzsqrt(a^2+b^2)tanalpha=0 .

The plane a x+b y=0 is rotated through an angle alpha about its line of intersection with the plane z=0. Show that he equation to the plane in the new position is ax+by±z sqrt(a ^ 2 +b^ 2) ​ tanα=0

The plane x-y-z=4 is rotated through an angle 90^(@) about its line of intersection with the plane x+y+2z=4 . Then the equation of the plane in its new position is

The plane x-2y+3z=0 is rotated through a right angle about the line of intersection with the plane 2x+3y-4z-5=0, find the equation of the plane in its new position.

The plane 4x+7y+4z+81=0 is rotated through a right angle about its line of intersection with the plane 5x+3y+10 z=25. The equation of the plane in its new position is x-4y+6z=k where k is

The plane x- y - z = 2 is rotated through an angle pi/2 about its line of intersection with the plane x + 2y + z = 2 . Find its equation in the new position.

The x-y plane is rotated about its line of intersection with the y-z plane by 45^0 , then the equation of the new plane is/are a. z+x=0 b. z-y=0 c. x+y+z=0 d. z-x=0