Parabola

Answer (a)

Step 1: Finding the normal equation Using differentiation for the hyperbola, Substituting , solving for , and forming the normal equation gives:

To solve the problem, we'll follow these steps: Step 1: Identify the focus of the parabola The given parabola is . The standard form of a parabola has its focus at . Comparing with , we get: Thus, the focus of the parabola is . Step 2: Parametrize a point on the parabola A general point on the parabola can be represented as , where is a parameter. This satisfies the equation: Step 3: Find the coordinates of The point divides the line segment joining the focus and the point on the parabola in the ratio . Using the section formula, the coordinates of are: Let . Then: Step 4: Eliminate the parameter From equation (2), solve for : Substitute into equation (1): Rearrange: Final Answer: The locus of is .

Step 1: Finding the hyperbola parameters The standard form of a hyperbola is: The angle between asymptotes is given by: Using , we find , and compute eccentricity . Step 2: Finding the pole The pole equation is determined by using the relation for pole with respect to hyperbola.

The given equation of the parabola is: Step 1: Finding the slope of the normal
Differentiating both sides with respect to : Solving for : At the point : The slope of the tangent at is , so the slope of the normal is the negative reciprocal: Step 2: Finding the equation of the normal
Using the point-slope form of a line equation: Substituting and : Expanding: Rearranging: Thus, the equation of the normal is:

Step 1: Finding the normals at given points The given equation of the parabola is: For points and , we use the normal equation for a parabola: Solving for the locus of intersection of the normals, we get:

Step 1: Finding the intersection points We substitute into the parabola equation : Solving this quadratic equation in gives the points and . Step 2: Finding the distance PQ Using the chord length formula: where (slope of line),

Step 1: Consider a chord of the hyperbola The equation of the hyperbola is: Let and be the endpoints of a chord. The midpoint of the chord is: Step 2: Condition that the chord touches the given parabola The equation of the given parabola is: The chords of the hyperbola that touch this parabola satisfy a special midpoint locus equation, which has been derived using midpoint properties and conic section relationships. Step 3: The required locus equation The locus of the midpoint of such chords is given by: Thus, the correct answer is option (A).

Step 1: Find the eccentricity of the ellipse For the ellipse: The eccentricity of an ellipse is given by: Here, , so: Step 2: Find the eccentricity of the hyperbola For the hyperbola: The eccentricity of a hyperbola is given by: Here, , and , so: Step 3: Using the given condition We are given that: Substituting the values: Squaring both sides: Multiplying by 144 to clear fractions: