Question

In a circle 2 chords PQ and RS intersects at O, which of the following is correct.

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

When you draw 2 lines from a point to a circle there are three possibilities,

The two lines are secants of the circle and intersect inside the circle (figure 1). In this case, we have.

AE.CE=BE.DE

One of the lines is tangent to the circle while the other is a secant (figure 2). In this case, we have.

$A{B}^2=BC.BD$

Both lines are secants of the circle and intersect outside of it (figure 3). In this case, we

have CB.CA=CD.CE

If you want to prove these relations draw extra lines to complete 2 similar triangles. then by using the properties of similar triangles these relations can be easily solved. This results are part of the power of a point theorem in which power of a point reflects the distance of a point from the circle.

Here $triangle$ ORQ and $△$ OPS are similar triangles. Hence

$\frac{OQ}{OS}=\frac{OR}{OP}=\frac{QR}{PS}$

$\Rightarrow OS\times OR=OP\times OQ$