A circle in the -plane has its center at . Line is tangent to this circle at the point , where is a constant. The slope of line is . What is the value of ?
Explanation
Question Type: Circle (Tangent Line)
💡 Trick: Tangent line ⊥ radius at point of tangency. Perpendicular slopes multiply to -1.
Solution:
- Center: (2, 9), tangent point: (a, -4)
- Slope of radius = (9 - (-4))/(2 - a) = 13/(2 - a)
- Slope of tangent = 6/5
- Perpendicular: 13/(2-a) × 6/5 = -1
- 78/(5(2-a)) = -1
- 78 = -5(2 - a)
- 78 = -10 + 5a
- 5a = 88
- a = 88/5
Answer: D) 88/5