Parabola

We have,






Vertex
Equations of directrix

The number of normals to a parabola that can be drawn through a given point is given by solving the equation of the normal to the parabola. In this case, solving for the number of normals through the point gives us 3 solutions. Thus, the correct answer is .

We are given the parabola: and the point on the parabola. We need to find where the normal at meets the parabola again at and compute: --- Step 1: Find Parameter for For the standard parabola , we compare with to get: The parametric form of the parabola: Substituting : Since , solving for : Thus, corresponds to . --- Step 2: Find Equation of the Normal at The equation of the normal to the parabola at the parametric point is: Substituting and : --- Step 3: Find the Second Intersection To find the second intersection, substitute into the parabola equation: Expanding: Solving for using the quadratic formula: Since is one root, the second intersection corresponds to: Finding : --- Step 4: Compute --- Final Answer: \bigskip

Step 1: Eliminate the radicals and rearrange.

Given the equation:

cross-multiply to obtain:

Next, square both sides (being careful) and move all terms to one side to form a polynomial equation in .

Step 2: Solve the resulting equation.

Expanding both sides, we get:

Simplify this expression and solve for . This process should yield two real solutions, though there may be extraneous solutions to check.

Step 3: Select the root in the interval .

Among the real solutions, determine which one falls between and . The correct root is .

To solve this problem, we start by recalling the equation of the hyperbola: We can write it in the standard form by dividing through by 144: This represents a hyperbola with a horizontal transverse axis. The asymptotes of this hyperbola are given by: The point is the intersection of the latus rectum and an asymptote. The equation for the latus rectum of a hyperbola is given by: Here, and , so the latus rectum equation becomes: Thus, the value of when and . Hence, the correct answer is .