Intercept of a Line or Length of a Chord

There are few things which student should know when dealing with online quiz. Here are some shortcuts which can lead to success.The length of a chord AB of a circle (or the intercept made by a circle on a line) is given by

AB=2√a^2-p^2

Where ‘a’ is a radius of the circle and ‘p’ is the length of the perpendicular from its centre on the chord. In particular the circle x^2+y^2+2gx+2fy+c=0 cuts

i. Intercept on x-axis=2√g^2-c

ii. Intercept on y-axis=2√f^2-c

EX: Find the length of intercept on y-axis, by a circle whose diameter is the line joining the point

(-4, 3) and (12,-1)

Solution: Here the equation of the circle (x+4) (x-12) + (y-3) (y+1) = 0

Or x^2+y^2-8x-2y-51=0

Here intercept on y-axis

= 2√f^2-c

=2√1-(-51)

=4√13

POSITION OF A POINT AND LINE WITH RESPECT TO A CIRCLE

Position of a point

A point (x, y) lies outside, on or inside a circle S= x^2 + y^2 + 2gx + 2fy + c = 0 according as

S=x^2 + 2gx + 2fy + c is positive, zero or negative. So

•  S > 0 =(x , y) is outside the circle 
•  S > 0 = (x , y) is on the circle 
•  S > 0 = (x , y) is inside the circle 

Position of a line 

Let L=0 be a line and S=0 be a circle. If ‘a’ be the radius of the circle and ‘p’ be the length of the perpendicular from its centre on the line, the

•  p > a = line is outside the circle 
•  p = a = line touches the circle 
•  p < a = line is a chord of the circle 
•  p = 0 = line is a diameter of the circle