and point A ( ω sin , ω cos p p on it. Therefore, by point-slope form, the equation of the line L is Fig . MATHEMATICS cos ω sin ω cos ω or cos ω sin ω ω ω ) sin cos sin ω p p p = − or x cos ω + y sin ω = p . Hence, the equation of the line having normal distance p from the origin and angle ω which the normal makes with the positive direction of x -axis is given by x cos ω + y sin ω = p ...
( ) Example Find the equation of the line whose perpendicular distance from the origin is units and the angle which the normal makes with positive direction of x -axis is °. Solution Here, we are given p = and ω = (Fig10. ). Now cos ° = and sin 15º = − (Why?) By the normal form ( ) above, the equation of the line is cos sin or or This is the required equation.
Example The Fahrenheit temperature F and absolute temperature K satisfy a linear equation. Given that K = when F = and that K = when F = . Express K in terms of F and find the value of F , when K = . Solution Assuming F along x -axis and K along y -axis, we have two points ( , ) and ( , ) in XY - plane.
By two-point form, the point (F, K) satisfies the equation K F or K F or K F ... ( ) which is the required relation. Fig . STRAIGHT LINES When K = , Equation ( ) gives F or F or F= .
× = − = − Alternate method We know that simplest form of the equation of a line is y