Question

If a circle with centre at $(-1, 1)$ touches the line $x + 2y + 4 = 0$, then the coordinates of the point of contact are \dots

• A. $(-2, -1)$
• B. $(8, -6)$
• C. $(-10, 3)$
• D. $(-2, -1)$ Note: The options provided contain a duplicate '(-2, -1)'. Selecting (a) is appropriate.

Hint:

Multiple choice shortcut: Instead of deriving the perpendicular line, simply plug the options into the line equation $x + 2y + 4 = 0$.
For $(-2, -1)$: $-2 + 2(-1) + 4 = -2 - 2 + 4 = 0$. It lies on the line!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A circle touches a straight line, making that line a tangent to the circle. The point of contact is geometrically defined as the foot of the perpendicular drawn exactly from the center of the circle to the tangent line.

Step 2: Key Formula or Approach:
We must find the intersection of two lines:
1. The given tangent line: $x + 2y + 4 = 0$.
2. A perpendicular line passing through the circle's center $(-1, 1)$.
The normal vector (slope components) of a line $Ax + By + C = 0$ is $(A, B)$. A line perpendicular to it has a direction vector $(A, B)$.

Step 3: Detailed Explanation:
The given line equation is $x + 2y + 4 = 0$. Its normal vector is $(1, 2)$.
The radius line (perpendicular to the tangent at the point of contact) must travel in the direction of this normal vector, starting from the center $(-1, 1)$.
We can write the parametric equations for this perpendicular radius line:
$$x = -1 + 1t$$
$$y = 1 + 2t$$
To find where this radius intersects the tangent line (the point of contact), substitute these parametric expressions directly into the tangent line's equation:
$$(-1 + t) + 2(1 + 2t) + 4 = 0$$
Expand and solve for $t$:
$$-1 + t + 2 + 4t + 4 = 0$$
$$5t + 5 = 0$$
$$5t = -5 \implies t = -1$$
Now, substitute $t = -1$ back into our parametric equations to get the $(x, y)$ coordinates of the point of contact:
$$x = -1 + (-1) = -2$$
$$y = 1 + 2(-1) = 1 - 2 = -1$$
The coordinates are $(-2, -1)$.

Step 4: Final Answer:
The point of contact is $(-2, -1)$, matching option (a).