If p is the 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)`.

If p is the length of the perpendicular from the origin on the line whose intercepts on the axes are a and b, then-

if P is the length of perpendicular from origin to the line x/a+y/b=1 then prove that 1/(a^2)+1/(b^2)=1/(p^2)

If p_(1)andp_(2) be the lenghts of the perpendiculars from the origin upon the lines 4x+3y=5cosalphaand 6x-8y=5sinalpha repectively , show that p_(1)^(2)+4p_(2)^(2)=1 .

If p and q are the lengths of perpendiculars from the origin to the lines xcostheta-ysintheta=kcos2theta and xsec""theta+y" cosec"""theta=""k , respectively, prove that p^2+4q^2=k^2 .

If the lengths of the perpendiculars from the vertices of a triangle ABC on the opposite sides are p_(1), p_(2), p_(3) then prove that (1)/(p_(1)) + (1)/(p_(2)) + (1)/(p_(3)) = (1)/(r) = (1)/(r_(1)) + (1)/(r_(2)) + (1)/(r_(3)) .

In the right-angled triangle ABC, /_C = 90^(@) .If the length of perpendicular drawn from C on AB be p and AB=c,BC =a,CA=b , then prove that (a)(1)/(p^(2))=(1)/(a^(2) )+ (1)/(b^(2)) , (b) pc = ab .

If p_(1) and p_(2) be the lengths of the perpendiculars from the origin upon the tangent and normal respectively to the curve x^((2)/(3)) +y^((2)/(3)) = a^((2)/(3)) at the point (x_(1), y_(1)) , then-

If P_(1)andP_(2) be the lenghts of the perpendiculars from the origin upon the lines xsintheta+ycostheta=(a)/(2)sin2thetaandxcostheta-ysintheta=acos2theta , prove that 4p_(1)^(2)+p_(2)^(2)=a^(2) .

If the perpendicular distance from origin to the straight line 4x-3y+P=0 is 2 unit,then the value of P is

If p_(1),p_(2) be the lenghts of perpendiculars from origin on the tangent and the curve x^((2)/(3))+y^((2)/(3))=a^((2)/(3)) drawn at any point on it, show that, 4p_(1)^(2)+p_(2)^(2)=a^(2)