Question

A circle of diameter \( R \) touches the parabola \( x^2 + y^2 - 4y = 0 \) and passes through the point \( (4, 5) \). Which of the following is correct?

• A. \( 3 \leq R \leq 7 \)
• B. \( 0<R<3 \)
• C. \( R>7 \)
• D. \( \frac{3}{2} \leq R \leq \frac{7}{2} \)

Hint:

Transform general conics into recognizable forms by completing the square. Use geometric constraints for tangency and point inclusion.

Answer 1

The Correct Option is A

Given that the circle:
- touches the parabola \( x^2 + y^2 = 4y \Rightarrow x^2 + (y - 2)^2 = 4 \) (circle with center \( (0, 2) \), radius 2)
- and passes through \( (4, 5) \)
Let the new circle have center \( (h, k) \), and radius \( R/2 \)
The condition that the circle touches the given parabola implies:
- The shortest distance from the center to the parabola must be equal to \( R/2 \)
Because the parabola is symmetric and resembles a circle \( x^2 + y^2 - 4y = 0 \), completing square gives: \[ x^2 + (y - 2)^2 = 4 \Rightarrow \text{Circle of radius 2} \] Now, any circle touching this one externally must have its center at a distance = sum of radii = \( 2 + R/2 \)
Also, the circle passes through \( (4, 5) \), giving constraints on \( R \)
Based on geometric computation or estimation, valid \( R \in [3, 7] \)