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 Concept:
The point of contact is the foot of the perpendicular from the center of the circle to the tangent line.

Step 2: Formula Application:
Foot of perpendicular $(h, k)$ from $(x_1, y_1)$ to $ax + by + c = 0$ is: $\frac{h - x_1}{a} = \frac{k - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2}$.

Step 3: Explanation:
$\frac{h - (-1)}{1} = \frac{k - 1}{2} = -\frac{1(-1) + 2(1) + 4}{1^2 + 2^2}$ $\frac{h+1}{1} = \frac{k-1}{2} = -\frac{-1+2+4}{5} = -\frac{5}{5} = -1$. $h+1 = -1 \implies h = -2$. $k-1 = -2 \implies k = -1$.

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