If a plane meets the equations axes at `A ,Ba n dC` such that the centroid of the triangle is `(1,2,4),` then find the equation of the plane.

If a plane meets the coordinate axes at A,B,C such that the centroid of the triangle ABC is the point (u,v,w), find the eqution of the plane.

The foot of the perpendicular drawn from the origin to a plane is (1,2,-3)dot Find the equation of the plane. or If O is the origin and the coordinates of P is (1,2,-3), then find the equation of the plane passing through P and perpendicular to O Pdot

A variable plane passes through a fixed point (alpha,beta,gamma) and meets the axes at A ,B ,a n dCdot show that the locus of the point of intersection of the planes through A ,Ba n dC parallel to the coordinate planes is alphax^(-1)+betay^(-1)+gammaz^(-1)=1.

The foot of the perpendicular drawn from the origin in a plane is (8, -4, -3) . Find the equation of the plane.

The equations of the perpendicular bisectors of the sides A Ba n dA C of triangle A B C are x-y+5=0 and x+2y=0 , respectively. If the point A is (1,-2) , then find the equation of the line B Cdot

A sphere of constant radius k , passes through the origin and meets the axes at A ,Ba n d Cdot Prove that the centroid of triangle A B C lies on the sphere 9(x^2+y^2+z^2)=4k^2dot

A variable plane l x+m y+n z=p(w h e r el ,m ,n are direction cosines of normal ) intersects the coordinate axes at points A ,Ba n dC , respectively. Show that the foot of the normal on the plane from the origin is the orthocenter of triangle A B C and hence find the coordinate of the circumcentre of triangle A B Cdot

If the middle points of the sides of a triangle are (-2,3),(4,-3),a n d(4,5) , then find the centroid of the triangle.

Consider the vector equation of two planes vecr(2hati+hatj+hatk)=3 and vecr(hati-hatj-hatk)=4 i.Find the vector equation of any plane through the intersection of the above two planes. ii. Find the vector equation of the plane through the intersection of the above two planes and the point (1,2,-1).