Parabola

The given parabola is: This is of the standard form where

The equation of the tangent to the parabola at point is:

We are told that the tangent passes through the external point , and makes an angle with the x-axis such that .

So, we are to find the point on the parabola such that the tangent from there:

1. Satisfies (since the point lies on the parabola)
2. Passes through
3. Has slope such that

Alternate method: Let us assume general equation of tangent in slope form for the parabola :

This tangent must pass through . So substitute: Multiply both sides by :

Solve the quadratic:

So:

Now use the condition . So choose:

Substitute this back into the slope form of tangent:

Find point of contact: For the parabola , and the tangent , find point of contact using standard method. Let the point be , then the tangent at is:

But we already know that the tangent is: Multiply both sides by : Try a better way — we already have the slope , so plug into: Let’s find point of contact on the parabola which lies on this tangent. Let point be Then: Substitute in first:

Multiply entire equation by 16: Solve:

Now find :

Answer:

We are given the parabola and the point , where is the parameter. The tangent and normal at any point on the parabola intersect the x-axis at the points and , respectively.

Step 1: Equation of the tangent
The general equation of the tangent at any point on the parabola is given by: For the point , the equation of the tangent becomes: Simplifying, we get the equation of the tangent as: This is the equation of the tangent.

Step 2: Finding the point on the x-axis
To find the x-intercept of the tangent (point ), set in the tangent equation: Solving for , we get: So, the point is .

Step 3: Equation of the normal
The equation of the normal at any point on the parabola is: For the point , the equation of the normal becomes: Simplifying, we get the equation of the normal: This is the equation of the normal.

Step 4: Finding the point on the x-axis
To find the x-intercept of the normal (point ), set in the normal equation: Solving for , we get: So, the point is .

Step 5: Finding the radius of the circle through , , and
The points , , and form a circle. The radius of the circle through these points is given by the formula: Substituting the coordinates of , , and , we get:

Answer:

We are given the parabola , and two focal chords and . The points are points on the parabola, and the line joining two points on a parabola is called a chord. A focal chord is a chord that passes through the focus of the parabola.

Step 1: Properties of Focal Chords
For the parabola , the focus is at , and the directrix is the line . One important property of the parabola is that the product of the slopes of any two points on a focal chord is constant. In fact, if and are points on a focal chord, the relationship between the coordinates of these points satisfies: This property holds for any pair of points on the focal chord.

Step 2: Intersection of Chords
The chords and intersect at a point that has a special geometric property: they intersect on the directrix of the parabola. This result is a well-known property of the geometry of parabolas and follows from the focus-directrix definition of the parabola.

Conclusion:
Therefore, the chords and intersect on the directrix of the parabola.

Answer:

Given: The parabola is:

Step 1: Compare with the standard form :
So, the focus of the parabola is:

Step 2: A ray of light is incident toward the focus at an angle where:
So, the slope of the incident ray is:
Using point-slope form, the equation of the incident ray passing through focus is:


Step 3: Let the point of incidence on the parabola be Since lies on the parabola , we have: (Equation 1)
Also, lies on the incident ray , so: (Equation 2)

Step 4: Substitute Equation 2 into Equation 1:





Solve this quadratic:
You get two roots, one positive and one negative. Keep the positive root: (since the other value is negative and not valid in the context)

Step 5: Now, use Equation 2 to find : But since we want the line along which the reflected ray travels, and: Property of parabola: Any ray directed at the focus reflects parallel to the axis
Axis of the parabola is the x-axis
So the reflected ray is horizontal and passes through point (corrected from earlier) Thus, equation of the reflected ray is:

Final Answer: (B):