The line joining the points A ( 0 , 5 ) and B ( 4 , 1 ) is a tangent to a circle whose centre C is at the point ( 4 , 4 ) . Find the equation of the line through C which is perpendicular to the line AB .

Equation of line AB , A ( 0 , 5 ) and B ( 4 , 1 ) y - y 1 y 2 - y 1 = x - x 1 x 2 - x 1 ⇒ y - 5 1 - 5 = x - 0 4 - 0 ⇒ 4 ( y - 5 ) = - 4 x gives y - 5 = - x . Hence, the required equation of the line A B is x + y - 5 = 0 . The equation of a line which is perpendicular to the line AB : x + y - 5 = 0 is x - y + k = 0 Since it is passing through the point ( 4 , 4 ) , we have 4 - 4 + k = 0 gives k = 0 The equation of a line which is perpendicular to AB and through C is x - y = 0

The line joining the points A ( 0 , 5 ) and B ( 4 , 1 ) is a tangent to a circle whose centre C is at the point ( 4 , 4 ) . Find the coordinates of the point of contact of line AB with the circle.

Equation of line AB , A ( 0 , 5 ) and B ( 4 , 1 ) y - y 1 y 2 - y 1 = x - x 1 x 2 - x 1 ⇒ y - 5 1 - 5 = x - 0 4 - 0 ⇒ 4 ( y - 5 ) = - 4 x gives y - 5 = - x . Hence, the required equation of the line A B is x + y - 5 = 0 . The equation of a line which is perpendicular to the line AB : x + y - 5 = 0 is x - y + k = 0 Since it is passing through the point ( 4 , 4 ) , we have 4 - 4 + k = 0 gives k = 0 The equation of a line which is perpendicular to AB and through C is x - y = 0 The coordinate of the point of contact P of the line AB with the circle is point of intersection of line. x + y - 5 = 0 and x - y = 0 solving, we get x = 5 2 and y = 5 2 Therefore, the coordinate of the point of contact is P 5 2 , 5 2 .

E M B I B E

Mathematics Standard 10>Chapter 5 - Coordinate Geometry>Solved Example

Tamil Nadu Board Solutions for Chapter: Coordinate Geometry, Exercise 37: Solved Example

Author:Tamil Nadu Board

Tamil Nadu Board Mathematics Solutions for Exercise - Tamil Nadu Board Solutions for Chapter: Coordinate Geometry, Exercise 37: Solved Example

Attempt the free practice questions on Chapter 5: Coordinate Geometry, Exercise 37: Solved Example with hints and solutions to strengthen your understanding. Mathematics Standard 10 solutions are prepared by Experienced Embibe Experts.