In a three-dimensional coordinate system, `P ,Q` , and `R`are images of a point `A(a ,b ,c)` in the `x-y ,y-z` and `z-x` planes, respectively. If `G` is the centroid of triangle `P Q R ,` then area of triangle `AOG` is (`O` is the origin) (A) `0` (B) `a^2+b^2+c^2` (C) `2/3(a^2+b^2+c^2)` (D) none of these

Show that a x+b y+r=0,b y+c z+p=0a n dc z+a x+q=0 are perpendicular to x-y ,y-za n dz-x planes, respectively.

Show that a x+b y+r=0,b y+c z+p=0a n dc z+a x+q=0 are perpendicular to x-y ,y-za n dz-x planes, respectively.

A plane meets the coordinates axes respectively at A, B and C such that the centroid of the triangle ABC is ( 3, 3, 3) . If the equation of the plane is x+y+z=p. Then p is equal to

(1, -2) and (-2, 3) are the points A and B of the DeltaABC . If the centroid of the triangle be at the origin, then find the coordinates of C.

The line x+y=p meets the x- and y-axes at Aa n dB , respectively. A triangle A P Q is inscribed in triangle O A B ,O being the origin, with right angle at QdotP and Q lie, respectively, on O Ba n dA B . If the area of triangle A P Q is 3/8t h of the are of triangle O A B , the (A Q)/(B Q) is equal to (a) 2 (b) 2/3 (c) 1/3 (d) 3

If the origin is the centroid of the triangle PQR with vertices P(2a, 2,6), Q(-4,3b,-10) & R(8,14,2c) then find the values of a, b & c

If A(1,p^2),B(0,1) and C(p ,0) are the coordinates of three points, then the value of p for which the area of triangle A B C is the minimum is (a) 1/(sqrt(3)) (b) -1/(sqrt(3)) (c) 1/(sqrt(2)) (d) none of these

A plane meets the coordinate axes at P, Q, R such that the centroid of the triangle PQR ar (a, b, c). If the equation of the plane is (x)/(a)+(y)/(b)+(z)/(c )=m , then the value of m is -

If tangents P Qa n dP R are drawn from a variable point P to thehyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,(a > b), so that the fourth vertex S of parallelogram P Q S R lies on the circumcircle of triangle P Q R , then the locus of P is (a) x^2+y^2=b^2 (b) x^2+y^2=a^2 (c) x^2+y^2=a^2-b^2 (d) none of these

A plane meets the coordinates axes in A, B, C such that the centroid of the triangle ABC is the point (a,a,a) .Then the equation of the plane is x+y+z=p , where p is ___