# Find the Points Where the Line Cuts the Circle

To find the points where the given line cuts the circle, for this we take an example as follows:

Example: Find the points where the line $3x + y - 5 = 0$ cuts the given circle ${x^2} + {y^2} = 25$. Also the length of the chord cut off form line by circle.
We have given line and the circle

First we find the points of intersection of line (i) and the given circle (ii), by using the method of solving simultaneous equations, from the line (i) we take the variable $y$ separate as follows

Putting the value of $y$ from (iii) in equation (ii) we get the results

Putting $x = 0$ in equation (iii), we have $y = 5$. This shows that one point of intersection of line and circle is $A\left( {0,5} \right)$. Next by putting $x = 3$ again in equation (iii) we have $y = - 4$. This shows that the other point of intersection of line and circle is $B\left( {3, - 4} \right)$.
The length of the chord cut off from the line and the circle is given as using distance formula applying on these two points