Question

If a circle with center \((-1,1)\) touches the line \(x+2y+4 = 0\), then the co-ordinates of the point of contact are

• A. \((-2,-1)\)
• B. \((8,-2)\)
• C. \((2,-3)\)
• D. \((4,-4)\)

Hint:

The point of contact is the foot of the perpendicular from the centre to the tangent line.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A tangent touches a circle at the foot of the perpendicular dropped from the centre to the line.

Step 2: Find the foot of the perpendicular.
For a line \(ax + by + c = 0\) and a point \((x_0, y_0)\), the foot \((x, y)\) satisfies \(\dfrac{x - x_0}{a} = \dfrac{y - y_0}{b} = -\dfrac{ax_0 + by_0 + c}{a^2 + b^2}\).

Step 3: Detailed Explanation:
Here \(a = 1, b = 2, c = 4\) and \((x_0, y_0) = (-1, 1)\). Then \(ax_0 + by_0 + c = -1 + 2 + 4 = 5\) and \(a^2 + b^2 = 5\), so the common ratio is \(-1\).
\[ x = -1 + 1(-1) = -2, \qquad y = 1 + 2(-1) = -1 \]

Step 4: Check.
The point \((-2, -1)\) lies on the line: \(-2 - 2 + 4 = 0\). The distance from the centre to this point is \(\sqrt{1 + 4} = \sqrt{5}\), which equals the radius \(\dfrac{|-1 + 2 + 4|}{\sqrt{5}} = \sqrt{5}\). So the circle touches the line at this point.

Final Answer:
The point of contact is \((-2, -1)\), option (A). \[ \boxed{(-2, -1)} \]