If the point C - 1 , 2 divides internally the line segment joining A 2 , 5 and B x , y in the ratio 3 : 4 , find the coordinates of B .

Let the coordinates of B are ( x , y ) It is given that the line segment divide in the ratio 3 : 4 So, m 1 = 3 and m 2 = 4 and x ' , y ' = ( - 1 , 2 ) ; x 1 , y 1 = ( 2 , 5 ) ; x 2 , y 2 = ( x , y ) Using section formula for the internal division, we get x ' , y ' = m 1 x 2 + m 2 x 1 m 1 + m 2 , m 1 y 2 + m 2 y 1 m 1 + m 2 …(i) ⇒ ( - 1 , 2 ) = 3 ( x ) + 4 ( 2 ) 3 + 4 , y = 3 ( y ) + 4 ( 5 ) 3 + 4 ⇒ - 1 = 3 x + 8 7 , 2 = 3 y + 20 7 ⇒ 3 x + 8 = - 7 and 3 y + 20 = 14 ⇒ 3 x = - 7 - 8 and 3 y = 14 - 20 ⇒ 3 x = - 15 and 3 y = - 6 ⇒ x = - 5 and y = - 2 Hence, the coordinates of B are ( - 5 , - 2 )

Find the ratio in which the segment joining the point 1 , - 3 and 4 , 5 is divided by x ‐ axis? Also, find the coordinates of this point of the x ‐ axis.

Let the ratio of intersection be l : m and the coordinates on x ‐ axis be x , 0 . We know that the coordinates of the point P x , y which divides the line segment joining the points x 1 , y 1 and x 2 , y 2 internally, in the ratio m 1 : m 2 is given x , y = m 2 x 1 + m 1 x 2 m 1 + m 2 , m 2 y 1 + m 1 y 2 m 1 + m 2 . According to the question: x , 0 = 4 l + m l + m , 5 l - 3 m l + m By comparing the ordinate we get, 5 l - 3 m l + m = 0 ⇒ 5 l - 3 m = 0 ⇒ 5 l = 3 m ⇒ l m = 3 5 ⇒ l : m = 3 : 5 Thus, l = 3 and m = 5 Now, x = 4 l + m l + m = 4 × 3 + 5 3 + 5 = 17 8 Hence, the required ratio is 3 : 5 and the coordinate of the point on the x ‐ axis is 17 8 , 0 .

Find the coordinates of a point A , where A B is diameter of a circle whose centre is 2 , - 3 and B is the point 1 , 4 .

Given, A B is diameter of a circle whose centre is P 2 , - 3 and B is the point 1 , 4 . We know that centre is the midpoint of a diameter. If x 1 , y 1 and x 2 , y 2 are two point then its midpoint is given by x , y = x 1 + x 2 2 , y 1 + y 2 2 . Let the coordinates of A be m , n . Therefore, m + 1 2 = 2 and n + 4 2 = - 3 ⇒ m = 4 - 1 and n = - 6 - 4 ⇒ m = 3 and n = - 10 Hence, the coordinates of A is 3 , - 10 .