If `p_(1) " and " p_(2)` are the lengths of the perpendiculars from the origin to the straight lines `xsectheta+ycosectheta=2a " and " xcostheta-ysintheta=acos2theta`, then prove that `p_(1)""^(2)+p_(2)""^(2)=a^(2)`.

If p and q are the lengths of perpendiculars from the origin to the lines x cos theta - y sin theta = k cos 2 theta " and " x sec theta + y cosec theta = k , respectively, prove that p^(2) + 4q^(2) = k^(2) .

If sintheta+costheta=p and sectheta+costheta=q , then prove that q(p^(2)-1)=2p

If sintheta+costheta=pandsectheta+cosectheta=q,"then prove that "q(p^(2)-1)=2p.

If p is the length of the perpendicular from the origin to the line whose intercepts will the coordinate axes are (1)/(3) and (1)/(4) then the value of p is

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 equation to the straight line passing through the point (acos^3theta,asin^3theta) and perpendicular to the line xsectheta+ycos e ctheta=a is (A) xcostheta-ysintheta=acos2theta (B) xcostheta+ysintheta=acos2theta (C) xsintheta+ycostheta=acos2theta (D) none of these

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))

P_(1) and P_(2) are the lengths of the perpendicular from the foci on the tangent of the ellipse and P_(3) and P_(4) are perpendiculars from extermities of major axis and P from the centre of the ellipse on the same tangent, then (P_(1)P_(2)-P^(2))/(P_(3)P_(4)-P^(2)) equals (where e is the eccentricity of the ellipse)

Find the coordinates of the foot of the perpendicular drawn from the point P(1,-2) on the line y = 2x +1. Also, find the image of P in the line.