Question

Two circles having radii r1 and r2 intersect orthogonally then the length of common chord is

Answer 1

The length of common chord is $\bf{h = \frac{2r_1r_2}{ \sqrt{r_1^2 + r_2^2}}}$

Step-by-step explanation:

Let us suppose h will be the length of the common chord.

Two circles with radii r1 and r2 are intersecting orthogonally means r1 through point of contact makes right angle with r2 through that point .

thus, the line intersecting has length. $= \sqrt{(r_1^2 + r_2^2)}$

Area of triangle formed, A1 $= \frac{1}{2}r_1r_2$

Area of triangle formed, A2 $= \frac{1}{2} \times \frac{h}{2} \times \sqrt{r_1^2+r_2^2$

Equating A1 = A2

$\frac{1}{2}r_1r_2 == \frac{1}{2} \times \frac{h}{2} \times \sqrt{r_1^2+r_2^2$

$h = \frac{2r_1r_2}{ \sqrt{r_1^2 + r_2^2}}$

Thus, the length of common chord is  $\bf{h = \frac{2r_1r_2}{ \sqrt{r_1^2 + r_2^2}}}$