• Coordinate Geometry
• OFFERED PRICE: Rs. 1,484
• View Details

Get extra Rs. 148 off

USE CODE: SELF10

Chord of Contact

 

Let AP and AQ be tangents to circle from point P(x1, y1). Then equation of PQ is known as equation of chord of contact. 

                    

For circle x2 + y2 = a2, it is 

x1x + y1y = a2 

For general circle, it is 

x1x + y1y + g(x1 + x) + f(y1 +y) + c = 0 



Note: 

1. It is also written as T = 0 

2. The equation of chord AB [A ≡ (R cos α, R sin α); B ≡ (R cos β, R sin β)] of the circle x2 + y2 = R2 is given by 

x cos ((α + β )/2) + y sin ((α - β )/2) = a cos ((α - β )/2) 

3. If a line y = mx + c intersects the circle x2 + y2 = a2 in two distinct points A and B then length of intercept AB = 2√((a2 (1+m2 )-c2)/(1+m2 )) 



Caution: 

The equation of a chord of contact and the equation of the tangent on a point of the circle and both given by T = 0. The difference is that while in the case of a tangent the point (x1, y1) lies on the circle. In the case of a chord of contact (x1, y1) lies outside the circle. 



Illustration: 

Write the equations of tangents to the circle x2 + y2 = 9 and having slope 2. 

Solution: 

Tangents with slope ‘m’ are given by y = mx + a √(1+m2 ) i.e. y = 2x ± 3 √(1+4). 



Note: 

These are two parallel tangents to the circle at the end of the diameter. 



Illustration: 

Write the equation of tangents to the circle x2 + y2 = 25 at the point (3, 4)? 

Solution: 

Point (3, 4) lies on the circle. 

Required equation of the tangent is : 3x + 4y= 25 using 

x1x + y1y = a2, where (x1, y1) ≡ (3, 4) 



Illustration: 

Write the equation of normal to x2 + y2 = 25 at (3, 4)? 

Solution: 

Recall: 

A normal to the circle passes through the centre of the circle. 

Normal is: y – 0 = 4/3 (x – 0) (using two point form of straight line) i.e. 3y – 4x = 0

To read more, Buy study materials of Circles comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.

• Complete JEE Main/Advanced Course and Test Series
• OFFERED PRICE: Rs. 15,900


• View Details Xpress Buy

Get Extra Rs. 1,590 off

COUPON CODE: SELF10

Course Features

• 731 Video Lectures
• Revision Notes
• Previous Year Papers
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Test paper with Video Solution