Conic Sections

Given:

Ellipse: , where

Foci:

Extremity of minor axis:

The lines joining the foci to the extremity of the minor axis are inclined at an angle of 60°. This implies the angle between the line joining (ae,0) and (0,b) to the x-axis is 30 degrees since 2 such lines create an isosceles triangle.

We want to find the eccentricity (e) of the ellipse.

Let's consider the focus (ae, 0) and the extremity of the minor axis (0, b). The slope of the line joining these two points is:

The angle between this line and the x-axis satisfies:

We are given that the angle between the line joining the focus and the endpoint of the minor axis is 60 degrees. Since the triangle formed by the focii and endpoint of minor axis is isoceles, the angle the line joining the focus to the x-axis is (180 - 60)/2 = 60 degrees. Therefore the other two angles are 60 degrees too.

So, , which means

We know that the relationship between a, b, and e in an ellipse is:

Substitute into this equation:

(Since eccentricity must be positive)

Therefore, the eccentricity of the ellipse is .

Given: Parabola with equation: Any point on this parabola can be represented as:

Let O = (0, 0) and point P divides the line segment OQ in the ratio 1:3 internally.

Using section formula:
Coordinates of P (h, k) are: So, P = (t, t^2/2)

Now eliminate parameter t:
From , we get .
Substitute into k:

Hence, the locus of point P is:

Final Answer: Option (D):