StudyGPT Just Leveled Up – Say Hello to Knowee! 🚀
Knowee
Questions
Features
Study Tools

If the normal at the end points of a variable chord PQ of parabola y2 – 4y – 2x = 0 are perpendicular, then the tangent at P & Q will intersect at

Question

If the normal at the end points of a variable chord PQ of parabola y24y2x=0 y^2 - 4y - 2x = 0 are perpendicular, then the tangent at P & Q will intersect at

🧐 Not the exact question you are looking for?Go ask a question

Solution

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.

This problem has been solved

Upgrade your grade with Knowee

Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.