Question

The circle on focal radii of a parabola as diameter touches

• A. the x-axis
• B. the tangent at the vertex
• C. the directrix
• D. None of the above

Hint:

When constructing circles with focal radii as diameters in conic sections, the circle will touch the tangent at the vertex of the parabola.

Answer 1

The Correct Option is B

Step 1: Understanding the parabola and its focal radii.
In a parabola, the focus is a fixed point from which distances to any point on the parabola are equal to the perpendicular distance from the directrix. The focal radius is the distance from the focus to a point on the parabola.

Step 2: Consider the geometry of the circle.
When a circle is constructed on the focal radii of the parabola as its diameter, we are essentially looking at a circle whose diameter is the distance from the focus to the point on the parabola along the tangent.

Step 3: Circle touches the tangent at the vertex.
The circle that uses the focal radii of the parabola as its diameter will touch the tangent at the vertex of the parabola. This is because the tangent at the vertex is perpendicular to the axis of symmetry of the parabola, and the circle touches it at exactly the point where the radius meets the vertex.

Step 4: Conclusion.
Thus, the correct answer is that the circle touches the tangent at the vertex, corresponding to option (B).