Circle And Parabola Geometry

We are given a point where and a parabola . The focus of this parabola is at . The shortest distance from the point to the parabola is given as 4. We are required to find the equation of a circle that passes through and the focus of the parabola, with its center on the axis of the parabola.

Step 1: Find the shortest distance to the parabola

We use the formula for the shortest distance from a point to the parabola :

Given that and the equation of the parabola is (where ), the distance is:

Thus, we have:

Since , we choose .

Step 2: Determine the center and equation of the circle

The center of the circle lies on the axis of the parabola, so it has coordinates . The circle passes through points and which is the focus of the parabola. The general equation of the circle is:

For the point :

For the point :

Equating the two expressions for , we have:

Solving this:

Substituting into the equation for :

The equation of the circle is therefore:

Expanding this, we get:

Step 3: Verify the options

Among the given options, the equivalent simplified equation is:

Thus, the correct option is:

Concept: For an ellipse centered at origin: The midpoint of a focal chord satisfies a standard locus relation derived from parametric representation of the ellipse.

Step 1: {Find semi-major axis.} Given:

Step 2: {Find .} Ellipse equation:

Step 3: {Use midpoint of focal chord property.} For an ellipse, the locus of midpoint of focal chords reduces to:

Step 1: Understanding the Concept:
We identify point A from the directrix, find point B using the given slope, determine point C as the other end of the focal chord, and then calculate the area of the triangle.
: Key Formula or Approach:
Parabola . Directrix: . Focal chord ends: and .
Step 2: Detailed Explanation:
For , . Focus . Directrix .
Point A (intersection of directrix and x-axis) is .
Let . Slope .
Given .
or .
If (Accepted).
If (Rejected).
So, . Since is a focal chord, .
.
Area of with :
Area .
Area .
Area .
Six times Area .
Step 3: Final Answer:
Six times the area of the triangle is 160.

Step 1: Understanding the Concept:
We need to estimate the range of values for and based on their arguments. We use the fact that and are monotonic and compare the given fractions to standard values (like , , etc.).
Step 2: Key Formula or Approach:
1. If , then .
2. If , then .
Step 3: Detailed Explanation:
For : . Thus . So, . Also, . Since is in the second quadrant, is negative. Statement II is true. For : . Thus . So, . Combining them: will likely be in the 3rd or 4th quadrant. Detailed estimation shows is actually negative. Statement I is false.
Step 4: Final Answer:
Statement I is false, but Statement II is true.