If the foot of the perpendicular from the origin to plane is `P(a ,b ,c)` , the equation of the plane is a. `x/a=y/b=z/c=3` b. `a x+b y+c z=3` c. `a x+b y+c z=a^(2)+b^2+c^2` d. `a x+b y+c z=a+b+c`

From the point P(a ,b ,c), let perpendicualars P La n dP M be drawn to Y O Za n dZ O X planes, respectively. Then the equation of the plane O L M is a. x/a+y/b+z/c=0 b. x/a+y/b-z/c=0 c. x/a-y/b-z/c=0 d. x/a-y/b+z/c=0

Let A B C be a triangle. Let A be the point (1,2),y=x be the perpendicular bisector of A B , and x-2y+1=0 be the angle bisector of /_C . If the equation of B C is given by a x+b y-5=0 , then the value of a+b is (a) 1 (b) 2 (c) 3 (d) 4

Solve the system of the equations: a x+b y+c z=d , a^2x+b^2y+c^2z=d^2 , a^3x+b^3y+c^3z=d^3 .

Let the equation of the plane containing the line x-y - z -4=0=x+y+ 2z-4 and is parallel to the line of intersection of the planes 2x + 3y +z=1 and x + 3y + 2z = 2 be x + Ay + Bz + C=0 Compute the value of |A+B+C| .

If the volume of tetrahedron A B C D is 1 cubic units, where A(0,1,2),B(-1,2,1)a n dC(1,2,1), then the locus of point D is a. x+y-z=3 b. y+z=6 c. y+z=0 d. y+z=-3

Find the equation of the plane containing the line (y)/(b)+(z)/(c)=1,x=0 , and parallel to the line (x)/(a)-(z)/(c)=1,y=0 .

A plane passes through a fixed point (a ,b ,c)dot Show that the locus of the foot of the perpendicular to it from the origin is the sphere x^2+y^2+z^2-a x-b y-c z=0.

Find the area of the triangle if (i) a=13, b=14, c=15 (ii) a:b :c = 3:4:5 and s=48 cm (iii) a=x/y + y/z, b = y/z + z/x and c=z/x + x/y

Consider the following linear equations: a x+b y+c z=0 b x+c y+a z=0 c x+a y+b z=0 Match the expression/statements in column I with expression/statements in Column II. Column I, Column II a+b+c!=0 and a^2+b^2+c^2=a b+b c+c a , p. the equations represent planes meeting only at a single point a+b+c=0a n da^2+b^2+c^2!=a b+b c+c a , q. the equations represent the line x=y=z a+b+c!=0a n da^2+b^2+c^2!=a b+b c+c a , r. the equations represent identical planes a+b+c!=0 and a^2+b^2+c^2!=a b+b c+c a , s. the equations represent the whole of the three dimensional space

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot