If the lines 2x - y = 0 and 2x - y = 5 are tangents to the circle, then find the diameter of the circle.

Steps to Solve

1. Identify the equations of the tangent lines

The given equations of the tangent lines are:

• Line 1: $2x - y = 0$ (which can be rearranged to $y = 2x$)
• Line 2: $2x - y = 5$ (which can be rearranged to $y = 2x - 5$)

2. Find the distance between the two parallel lines

Since both lines are parallel (same slope), we can use the formula to find the distance between two parallel lines given by $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$. The formula for the distance $d$ between these two lines is:

$$ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} $$

Here:

• Line 1 is $2x - y + 0 = 0$ ($C_1 = 0$)
• Line 2 is $2x - y + 5 = 0$ ($C_2 = 5$)
• Thus, $A = 2$, $B = -1$.

Substitute these values into the distance formula: $$ d = \frac{|5 - 0|}{\sqrt{2^2 + (-1)^2}} = \frac{5}{\sqrt{4 + 1}} = \frac{5}{\sqrt{5}} = \sqrt{5} \cdot 1 = \sqrt{5} $$

3. Relate the distance to the circle’s diameter

Knowing that the distance between the tangent lines is equal to the diameter of the circle (since each line is a tangent at a point that is the same distance from the center of the circle), we find:

$$ \text{Diameter} = d = \sqrt{5} $$