Question

Let \( S \) be the circumcircle of the triangle formed by the line \( x - 2y - 4 = 0 \) with the coordinate axes. If \( P(-2, -4) \) is a point in the plane of the circle \( S \) and \( Q \) is a point on \( S \) such that the distance between \( P \) and \( Q \) is the least, then \( PQ = \)

• A. \( 5 - \sqrt{5} \)
• B. \( 5 + \sqrt{5} \)
• C. \( 13 + \sqrt{5} \)
• D. \( 13 - \sqrt{5} \)

Hint:

When finding the shortest distance from a point to a circle, always use the perpendicular distance formula.

Answer 1

The Correct Option is A

The problem involves finding the shortest distance between the point \( P(-2, -4) \) and the circle \( S \). The shortest distance from a point to a circle is along the line connecting the point to the center of the circle, which is the perpendicular distance.
To solve for \( PQ \), we can use the formula for the distance from a point to a circle, knowing that the center and radius of the circle can be derived from the equation of the circumcircle.
The distance formula for \( PQ \) gives \( 5 - \sqrt{5} \) as the shortest distance. Therefore, the correct answer is \( 5 - \sqrt{5} \).