If A (1, -1),B (0, 4),C (-5, 3) are vertices of a triangle then find the slope of each side.
A (1, -1),B (0, 4),C (-5, 3) are the given points.Slope of line AB = (y2-y1)/ (x2-x1)= (4-(-1))/(0-1)= 5/-1 = -5Slope of line BC = (y2-y1)/ (x2-x1)= (3-4)/(-5-0)= -1/-5 = 1/5Slope of line AC = (y2-y1)/ (x2-x1)= (3-(-1))/(-5-1)= 4/-6 = -2/3Hence the slopes of the sides AB, BC and AC are -5, 1/5 , -2/3 […]
Determine whether the following points are collinear.
(1) A(-1, -1), B(0, 1), C(1, 3) are the given points.Slope of line AB = (y2-y1)/ (x2-x1)= (1-(-1))/(0-(-1))= 2/1 = 2Slope of line BC = (y2-y1)/ (x2-x1)= (3-1)/(1-0)= 2Slope of line AB and BC are equal.Point B lies on both lines.Point A,B,C are collinear. (2) D(-2, -3), E(1, 0), F(2, 1) are the given points.Slope […]
Find the slopes of the lines passing through the given points.
(1) Given A(2,3) and B(4,7)x1 = 2y1 = 3×2 = 4y2 = 7Slope of the line AB = (y2-y1)/ (x2-x1)= (7-3)/(4-2)= 4/2 = 2Hence Slope of line AB is 2. (2) Given P(-3,1) and Q(5,-2)x1 = -3y1 = 1×2 = 5y2 = -2Slope of the line PQ = (y2-y1)/ (x2-x1)= (-2-1)/(5-(-3))= -3/8Hence Slope of line PQ is -3/8. (3) Given C(5,-2) and D(7,3)x1 = […]
Angles made by the line with the positive direction of X-axis are given. Find the slope of these lines.
(1) Given angle made by line with positive direction of X axis, = 45˚.Slope of the line ,m = tanm = tan 45˚ = 1Hence slope of the line is 1. (2) Given angle made by line with positive direction of X axis, = 60˚.Slope of the line ,m = tanm = tan 60˚ = […]
Find the coordinates of midpoint of the segment joining the points (22, 20) and (0, 16).
Let (x1,y1) = (22,20)(x2,y2) = (0,16)Let co-ordinate of midpoint be A(x,y)By Midpoint formula x = (x1+x2)/2 and y = (y1+y2)/2x = (22+0)/2 = 11y = (20+16)/2 = 36/2 = 18Hence co-ordinates of midpoint are (11,18).
5. Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k .
Let P (x, y) , A (x1, y1) and B (x2, y2) and be the given points.Here, x = k, y = 7, x1 = 8, y1 = 9, x2 = 1, y2 = 2,By Section formula y = (my2+ny1)/(m+n)7 = m×2+n×9/(m+n)7 = 2m+9n/(m+n)7m+7n = 2m+9n7m-2n = 9n-7n5m = 2nm/n = 2/5Hence m:n = 2:5.By Section formula, x = […]
Point P is the centre of the circle and AB is a diameter . Find the coordinates of point B if coordinates of point A and P are (2, -3) and (-2, 0) respectively.
Given co-ordinates of P = (-2,0) = (x,y)Co-ordinates of A = (2,-3) = (x1, y1)Let co-ordinates of B = (x2,y2)Since P is the midpoint of diameter AB ,By midpoint formula x = (x1+x2)/2-2 = 2+x2/2×2 = -4-2×2 = -6y = (y1+y2)/20 = (-3+y2)/2-3+y2 = 0y2 = 3Hence the co-ordinates of point B is (-6,3).
Find the ratio in which point T(-1, 6)divides the line segment joining the points P(-3, 10) and Q(6, -8).
Let, point T(-1,6) divides segment PQ in the ratio m:nGiven P(-3, 10) and Q(6, -8).x = -1 , y = 6×1 = -3 , y1 = 10×2 = 6 , y2 = -8By Section formula, x = (mx2+nx1)/(m+n)-1 = m×6+n×-3/(m+n)-1 = 6m-3n/(m+n)-m+-n = 6m-3n2n = 7mm/n = 2/7m:n = 2:7Point T divides PQ is the ratio 2:7.
In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a:b.
(1) Let the co-ordinates of A be (x, y).P(-3, 7) and Q(1, -4) are the given points.x1 = -3 , y1 = 7 , x2 = 1 , y2 = -4 , m = 2 and n = 1By Section formula x = (mx2+nx1)/(m+n)x = (2×1+1×-3)/(2+1)x = -1/3By Section formula y = (my2+ny1)/(m+n)y = (2×-4+1×7)/3y = (-8+7)/3y = -1/3Hence […]
Find the coordinates of point P if P divides the line segment joining the points A(-1,7) and B(4,-3) in the ratio 2:3.
Let the co-ordinates of P be (x, y).A(-1,7) and B(4,-3) are the given points.x1 = -1 , y1 = 7 , x2 = 4 , y2 = -3 , m = 2 and n = 3By Section formula x = (mx2+nx1)/(m+n)x = (2×4+3×-1)/(2+3)x = (8-3)/5x = 5/5x= 1By Section formula y = (my2+ny1)/(m+n)y = (2×-3+3×7)/5y = (-6+21)/5y = 15/5y […]