Question

Let the shortest distance from \( (a, 0) \), where \( a > 0 \), to the parabola \( y^2 = 4x \) be 4. Then the equation of the circle passing through the point \( (a, 0) \) and the focus of the parabola, and having its center on the axis of the parabola is:

• A. \( x^2 + y^2 - 6x + 5 = 0 \)
• B. \( x^2 + y^2 - 4x + 3 = 0 \)
• C. \( x^2 + y^2 - 10x + 9 = 0 \)
• D. \( x^2 + y^2 - 8x + 7 = 0 \)

Hint:

To solve geometry problems involving circles and parabolas, first calculate the focus of the parabola and use symmetry of the problem to find the equation of the circle passing through given points.

Answer 1

The Correct Option is A

The equation of the given parabola is: \[ y^2 = 4x. \] This is of the form \( y^2 = 4ax \), where \( a = 1 \). Therefore, the focus of the parabola is at \( (1, 0) \). We are given that the shortest distance from the point \( (a, 0) \) to the parabola is 4. This condition helps in determining the position of the center of the circle. Next, the equation of the circle passing through the point \( (a, 0) \) and the focus \( (1, 0) \) and having its center on the axis of the parabola is given by: \[ (x - h)^2 + y^2 = r^2, \] where \( h \) is the center's x-coordinate and \( r \) is the radius. By solving using the distances from the center to the points \( (a, 0) \) and \( (1, 0) \), we find that the equation of the circle is: \[ x^2 + y^2 - 6x + 5 = 0. \]