Prove that if a plane has the intercepts a,b,c and is at a distance of p units from the origin , then `1/a^2+1/b^2+1/c^2=1/p^2`

Two systems of rectangular axes have the same origin. If a plane cuts them at distance a ,b ,c and a^prime ,b^(prime),c ' from the origin, then a. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0 b. 1/(a^2)-1/(b^2)-1/(c^2)+1/(a^('2))-1/(b^('2))-1/(c^('2))=0 c. 1/(a^2)+1/(b^2)+1/(c^2)-1/(a^('2))-1/(b^('2))-1/(c^('2))=0 d. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0

If the perpendicular distance of a point A, other than the origin from the plane x + y + z = p is equal to the distance of the plane from the origin, then the coordinates of A are (A) (p,2p,0) (B) (0,2p,-p) (C) (2p,p,-p) (D) (2p,-p,2p)

At a point P on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 tangents PQ is drawn. If the point Q be at a distance (1)/(p) from the point P, where 'p' is distance of the tangent from the origin, then the locus of the point Q is

If p_(1),p_(2),p_(3) are the altitudes of a triangle from the vertieces A,B,C and Delta is the area of the triangle then prove that (1)/(p_(1))+(1)/(p_(2))-(1)/(p_(3))=(2ab)/((a+b+c)Delta)"cos"^(2)(C)/(2)

If p is length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that 1/p^(2)=1/a^(2)+1/b^(2) .

The line L has intercepts a and b on the coordinate axes. Keeping the origin fixed, the coordinate axes are rotated through a fixed angle. If the line L has intercepts p and q on the rotated axes, then (1)/(a^(2))+(1)/(b^(2))  is equal to

If a ,b ,a n dc are in G.P. then prove that 1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)dot

The centres of two circles C_(1) and C_(2) each of unit radius are at a distance of 6 unit from each other. Let P be the mid-point of the line segment joining the centres of C_(1) and C_(2) and C be a circle touching circles C_(1) and C_(2) externally. If a common tangent to C_(1) and C passing through P is also a common tangent to C_(2) and C, then the radius of the circle C, is

In R^(3) , consider the planes P_(1):y=0 and P_(2),x+z=1. Let P_(3) be a plane, different from P_(1) and P_(2) which passes through the intersection of P_(1) and P_(2) , If the distance of the point (0,1,0) from P_(3) is 1 and the distance of a point (alpha,beta,gamma) from P_(3) is 2, then which of the following relation(s) is/are true?

If p_(2),p_(2),p_(3) are the perpendiculars from the vertices of a triangle to the opposite sides, then prove that p_(1)p_(2)p_(3)=(a^(2)b^(2)c^(2))/(8R^(3))