ncert math solution

Ncert misc straight line…..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.Let y = mx + c be the line through point (–1, 2).Accordingly, 2 = m (–1) + c.⇒ 2 = –m + c⇒ c = m + 2∴ y = mx + m + 2 … (1)The given line isx + y = 4 … (2)On solving equations (1) and (2), we obtain is the point of intersection of lines (1) and (2).Since this point is at a distance of 3 units from point (– 1, 2), according to distance formula,Thus, the slope of the required line must be zero i.e., the line must be parallel to the x-axis.If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value ofThe equations of the given lines arey = 3x + 1 … (1)2y = x + 3 … (2)y = mx + 4 … (3)Slope of line (1), m1 = 3Slope of line (2), Slope of line (3), m3 = mIt is given that lines (1) and (2) are equally inclined to line (3). This means thatthe angle between lines (1) and (3) equals the angle between lines (2) and (3).Thus, the required value of m is.If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y – 5 = 0 and 3x– 2y + 7 = 0 is always 10. Show that P must move on a lineThe equations of the given lines arex + y – 5 = 0 … (1)3x – 2y + 7 = 0 … (2)The perpendicular distances of P (x, y) from lines (1) and (2) are respectively given byIt is given that., which is the equation of a line.Similarly, we can obtain the equation of line for any signs of.Thus, point P must move on a line.The equations of the given lines are9x + 6y – 7 = 0 … (1)3x + 2y + 6 = 0 … (2)Let P (h, k) be the arbitrary point that is equidistant from lines (1) and (2). The perpendicular distance of P (h, k) from line (1) is given byThe perpendicular distance of P (h, k) from line (2) is given bySince P (h, k) is equidistant from lines (1) and (2), ∴9h + 6k – 7 = – 9h – 6k – 18⇒ 18h + 12k + 11 = 0Thus, the required equation of the line is 18x + 12y + 11 = 0.