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.

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 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 1/(sqrt(3)) (b) -1/(sqrt(3)) 1/(sqrt(2)) (d) none of these

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 x^2+y^2=b^2 (b) x^2+y^2=a^2 x^2+y^2=a^2-b^2 (d) none of these

Let P and Q be any two points on the lines represented by 2x-3y = 0 and 2x + 3y = 0 respectively. If the area of triangle OPQ (where O is origin) is 5, then which of the following is not the possible equation of the locus of mid-point of PO? (a) 4x^2-9y^2 +30 = 0 (b) 4x^2-9y^2-30 = 0 (c) 9x^2-4y^2-30=0 (d) none of these

If the plane x/2+y/3+z/6=1 cuts the axes of coordinates at points, A ,B ," and "C , then find the area of the triangle A B C

P is a point on the line y+2x=1, and Q and R two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these

A point P moves on line 2x-3y+4=0 If Q(1,4) and R(3,-2) are fixed points, then the locus of the centroid of triangle PQR is a line: (a) with slope 3/2 (b) parallel to y-axis (c) with slope 2/3 (d) parallel to x-axis

If a ,b , a n dc are in G.P. and x ,y , respectively, are the arithmetic means between a ,b ,a n db ,c , then the value of a/x+c/y is 1 b. 2 c. 1//2 d. none of these