To find the point of intersection of the tangents at points P and Q on the parabola, we need to follow these steps:

Step 1: Find the coordinates of points P and Q.
To find the coordinates of P and Q, we need to solve the equation of the parabola y^2 - 4y - 2x = 0 for the variable chord PQ. Let's assume the coordinates of P are (x1, y1) and the coordinates of Q are (x2, y2).

Step 2: Find the slope of the normal at point P.
The slope of the normal at point P can be found by taking the negative reciprocal of the derivative of the equation of the parabola at point P. The derivative of y^2 - 4y - 2x = 0 is given by dy/dx = -2/(2y - 4). Therefore, the slope of the normal at point P is -1/(dy/dx).

Step 3: Find the slope of the normal at point Q.
Similarly, the slope of the normal at point Q can be found by taking the negative reciprocal of the derivative of the equation of the parabola at point Q. The derivative of y^2 - 4y - 2x = 0 is given by dy/dx = -2/(2y - 4). Therefore, the slope of the normal at point Q is -1/(dy/dx).

Step 4: Check if the slopes of the normals are perpendicular.
To check if the slopes of the normals at points P and Q are perpendicular, we need to multiply the slopes and check if the product is -1. If the product is -1, then the normals are perpendicular.

Step 5: Find the equation of the tangent at point P.
The equation of the tangent at point P can be found by substituting the coordinates of P into the equation of the parabola y^2 - 4y - 2x = 0. This will give us an equation in terms of x and y, which represents the tangent at point P.

Step 6: Find the equation of the tangent at point Q.
Similarly, the equation of the tangent at point Q can be found by substituting the coordinates of Q into the equation of the parabola y^2 - 4y - 2x = 0. This will give us an equation in terms of x and y, which represents the tangent at point Q.

Step 7: Find the point of intersection of the tangents.
To find the point of intersection of the tangents, we need to solve the equations of the tangents obtained in steps 5 and 6 simultaneously. This will give us the coordinates of the point of intersection.

By following these steps, we can determine the point of intersection of the tangents at points P and Q on the parabola.

Question

To find the point of intersection of the tangents at points P and Q on the parabola, we need to follow these steps:

Step 1: Find the coordinates of points P and Q.
To find the coordinates of P and Q, we need to solve the equation of the parabola y^2 - 4y - 2x = 0 for the variable chord PQ. Let's assume the coordinates of P are (x1, y1) and the coordinates of Q are (x2, y2).

Step 2: Find the slope of the normal at point P.
The slope of the normal at point P can be found by taking the negative reciprocal of the derivative of the equation of the parabola at point P. The derivative of y^2 - 4y - 2x = 0 is given by dy/dx = -2/(2y - 4). Therefore, the slope of the normal at point P is -1/(dy/dx).

Step 3: Find the slope of the normal at point Q.
Similarly, the slope of the normal at point Q can be found by taking the negative reciprocal of the derivative of the equation of the parabola at point Q. The derivative of y^2 - 4y - 2x = 0 is given by dy/dx = -2/(2y - 4). Therefore, the slope of the normal at point Q is -1/(dy/dx).

Step 4: Check if the slopes of the normals are perpendicular.
To check if the slopes of the normals at points P and Q are perpendicular, we need to multiply the slopes and check if the product is -1. If the product is -1, then the normals are perpendicular.

Step 5: Find the equation of the tangent at point P.
The equation of the tangent at point P can be found by substituting the coordinates of P into the equation of the parabola y^2 - 4y - 2x = 0. This will give us an equation in terms of x and y, which represents the tangent at point P.

Step 6: Find the equation of the tangent at point Q.
Similarly, the equation of the tangent at point Q can be found by substituting the coordinates of Q into the equation of the parabola y^2 - 4y - 2x = 0. This will give us an equation in terms of x and y, which represents the tangent at point Q.

Step 7: Find the point of intersection of the tangents.
To find the point of intersection of the tangents, we need to solve the equations of the tangents obtained in steps 5 and 6 simultaneously. This will give us the coordinates of the point of intersection.

By following these steps, we can determine the point of intersection of the tangents at points P and Q on the parabola.

Knowee AI · Accepted Answer