Length of tendon in circle

Question

What is the length of chord that pass on two specific point.

For example I have circle ( r=1) point1 :(x1,y1) point2(x2,y2); length of chord?

Answer 1

here is how you will do. any point on the line connecting $(x_1, y_1)$ and $(x_2, y_2)$ can be written in the form $$x = tx_2 + (1-t)x_1, y = ty_2 + (1-t)y_1$$ for the point $(x,y)$ to be on the unit circle, $t$ needs to satisfy $$ \left(tx_2 + (1-t)x_1 \right)^2+ \left( ty_2 + (1-t)y_1\right)^2 = 1$$

follow these steps:

(a) solve the quadratic equation for $t$

(b) find the two roots $t_1, t_2$

(c) length of the chord is $|t_1 - t_2|\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$