Tangent to a Circle from a Line?

📌 CONCEPT: A tangent to a circle from a line can be formed when the line intersects the circle at exactly one point, resulting in a single point of tangency.

📐 RULE / FORMULA: The distance between the center of the circle and the point of tangency is equal to the radius of the circle.

💡 WORKED EXAMPLE: Consider a circle with center (0, 7) and radius 10. If the line y = 2x + 3 intersects the circle, we can find the points of tangency by equating the equation of the circle and the line. The equation of the circle is x^2 + (y-7)^2 = 100. By substituting y = 2x + 3 into the equation of the circle, we can solve for the points of tangency. If the line intersects the circle at exactly one point, it will be a tangent to the circle.

⚠️ COMMON MISTAKE: Students often mistakenly assume that the line will intersect the circle at two points if the distance between the center of the circle and the line is greater than the radius of the circle. However, the line can still intersect the circle at exactly one point if the distance between the center of the circle and the line is equal to the radius of the circle.