Distance of the straight line 4x+3y=5 `cosalpha`and 6x-8y=`5 sinalpha`from the origin are `p_1` and `p_2` respectivley show that `p_1^2+4p_2^2=1`

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 .

Find the perpendicular distance of the straight line 4x-y=5 from the point (2,-1)

If the straight line xcosalpha+ysinalpha=p touches the curve x y=a^2, then prove that p^2=4a^2cosalphasinalphadot

A straight lines L through the origin meets the lines x+y=1 and x+y=3 at P and Q respectively. Through P and Q two straight lines L_1 and L_2 are drawn, parallel to 2x-y=5 and 3x+y=5 respectively. Line L_1 and L_2 intersect at R. Show that the locus of R as L varies is a straight line.

If the straight line y= 4x -5 touches the curve y^(2) = px^(3) + q at (2, 3), then the values of p and q are-

Show that the equation of the straight line xcosalpha+ysinalpha=p can be expressed in the following form: (x-pcosalpha)/(-sinalpha)=(y-p sin alpha)/(cosalpha)=r

A straight line through the origin 'O' meets the parallel lines 4x +2y= 9 and 2x +y=-6 at points P and Q respectively. Then the point 'O' divides the segment PQ in the ratio : (A) 1:2 (B) 3:2 (C) 2:1 D) 4:3

If the straight lines 2x-3y+5=0andpx+2y=6 be parallel to each other , state which of the following is the value of p ,

A line intersects the straight lines 5x-y-4=0 and 3x-4y-4=0 at A and B , respectively. If a point P(1,5) on the line A B is such that A P: P B=2:1 (internally), find point Adot

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-