if the plane 2ax 3ay 4az 6 0 passes through the midpoint of

If the plane $$2ax - 3ay + 4az + 6 = 0$$ passes through the midpoint of the line joining the centres of the spheres $${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$$ and $${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$$ then $$a$$ equals :

A. $$ - 1$$

B. $$1$$

C. $$ - 2$$

D. $$2$$

Answer : $$ - 2$$

Solution :
Plane $$2ax - 3ay + 4az + 6 = 0$$ passes through the mid point of the centre of spheres $${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$$ and $${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$$ respectively. Center of spheres are $$\left( { - 3,\,4,\,1} \right)$$ and $$\left( {5,\, - 2,\,1} \right).$$
Mid point of centres is $$\left( {1,\,1,\,1} \right).$$
Satisfying this in the equation of plane, we get
$$2a - 3a + 4a + 6 = 0 \Rightarrow a = - 2.$$

Releted MCQ Question on
Geometry >> Three Dimensional Geometry

Releted Question 1

The value of $$k$$ such that $$\frac{{x - 4}}{1} = \frac{{y - 2}}{1} = \frac{{z - k}}{2}$$ lies in the plane $$2x - 4y + z = 7,$$ is :

A. $$7$$

B. $$ - 7$$

C. no real value

D. $$4$$

View Answer

Releted Question 2

If the lines $$\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{4}$$ and $$\frac{{x - 3}}{1} = \frac{{y - k}}{2} = \frac{z}{1}$$ intersect, then the value of $$k$$ is :

A. $$\frac{3}{2}$$

B. $$\frac{9}{2}$$

C. $$ - \frac{2}{9}$$

D. $$ - \frac{3}{2}$$

View Answer

Releted Question 3

A plane which is perpendicular to two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4,$$ passes through $$\left( {1,\, - 2,\,1} \right).$$ The distance of the plane from the point $$\left( {1,\,2,\,2} \right)$$ is :

A. $$0$$

B. $$1$$

C. $$\sqrt 2 $$

D. $$2\sqrt 2 $$

View Answer

Releted Question 4

Let $$P\left( {3,\,2,\,6} \right)$$ be a point in space and $$Q$$ be a point on the line $$\vec r = \left( {\hat i - \hat j + 2\hat k} \right) + \mu \left( { - 3\hat i + \hat j + 5\hat k} \right)$$
Then the value of $$\mu $$ for which the vector $$\overrightarrow {PQ} $$ is parallel to the plane $$x-4y+3z=1$$ is :

A. $$\frac{1}{4}$$

B. $$ - \frac{1}{4}$$

C. $$\frac{1}{8}$$

D. $$ - \frac{1}{8}$$

View Answer

If the plane $$2ax - 3ay + 4az + 6 = 0$$ passes through the midpoint of the line joining the centres of the spheres $${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$$ and $${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$$ then $$a$$ equals :

Releted MCQ Question on
Geometry >> Three Dimensional Geometry

The value of $$k$$ such that $$\frac{{x - 4}}{1} = \frac{{y - 2}}{1} = \frac{{z - k}}{2}$$ lies in the plane $$2x - 4y + z = 7,$$ is :

If the lines $$\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{4}$$ and $$\frac{{x - 3}}{1} = \frac{{y - k}}{2} = \frac{z}{1}$$ intersect, then the value of $$k$$ is :

A plane which is perpendicular to two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4,$$ passes through $$\left( {1,\, - 2,\,1} \right).$$ The distance of the plane from the point $$\left( {1,\,2,\,2} \right)$$ is :

Practice More Releted MCQ Question on
Three Dimensional Geometry

Practice More MCQ Question on Maths Section

If the plane $$2ax - 3ay + 4az + 6 = 0$$ passes through the midpoint of the line joining the centres of the spheres $${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$$ and $${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$$ then $$a$$ equals :

Releted MCQ Question on Geometry >> Three Dimensional Geometry

The value of $$k$$ such that $$\frac{{x - 4}}{1} = \frac{{y - 2}}{1} = \frac{{z - k}}{2}$$ lies in the plane $$2x - 4y + z = 7,$$ is :

If the lines $$\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 1}}{4}$$ and $$\frac{{x - 3}}{1} = \frac{{y - k}}{2} = \frac{z}{1}$$ intersect, then the value of $$k$$ is :

A plane which is perpendicular to two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4,$$ passes through $$\left( {1,\, - 2,\,1} \right).$$ The distance of the plane from the point $$\left( {1,\,2,\,2} \right)$$ is :

Practice More Releted MCQ Question on Three Dimensional Geometry

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Geometry >> Three Dimensional Geometry

Practice More Releted MCQ Question on
Three Dimensional Geometry