Straight Lines

If p and q are the lengths of perpendiculars from the origin to the lines x cosθ − y sinθ = k cos2θ and x secθ + y cscθ = k, prove that p² + 4q² = k².

Find perpendicular distance from the origin of the line joining the points (cosθ, sinθ) and (cosφ, sinφ).

Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line x – 2y = 3. 

Show that the equation of a line passing through the origin and making an angle θ with the line y = mx + c is

^y/_x = ±^{(m + tan θ)}/_{(1 - m tan θ)}

Find the point to which the origin should be shifted after shifting of origin so that the equation x² - 12x + 4 = 0 will have no first degree term. 

Find the direction in which a straight line must be drawn through the point (-1, 2) so that its point of intersection with the line x + y= 4 may be at a distance of 3 units from this point. 

If the slope of a line passing through the point A(3, 2) is 3/4 then find points on the line which are 5 units away from the point A. 

Find the points on the x-axis whose perpendicular distance from the straight line  x/a + y/b = 1  is a. 

Find the equation of line parallel to the y-axis and drawn through the point of intersection of x – 7y + 5 = 0 and 3x + y – 7 = 0

Find the new coordinates of point (3, –4) if the origin is shifted to (1, 2) by a translation.

Find the transformed equation of the straight line 2x – 3y + 5 + 0 when the origin is shifted to the point (3, –1) after translation of axes.