To determine the value of x that makes line segment rq tangent to circle p at point q, we need to use the properties of tangents and the geometry of circles. A tangent to a circle at a given point is perpendicular to the radius drawn to that point.
Assuming we have the radius of circle p and the coordinates of point q, we can set up an equation based on the distance formula and the slope of the line.
The first step is to find the slope of the radius (let’s denote its endpoints as the center of the circle, O, and point Q on the circle). If the radius has a slope m, then the slope of line rq must be the negative reciprocal of m because they are perpendicular.
Next, we can write the equation of line rq using its slope and one of the points through which it passes. Substitute the known coordinates and the derived slope into the equation. Set the equation equal to the equation of the circle to find the point of intersection.
Setting the quadratic equation formed by this intersection equal to zero will help in finding the x-values that yield real solutions. From there, the specific value of x that allows rq to be tangent (having exactly one solution) can be established.
Once you solve for x, you can confirm its validity by plugging it back into the original tangent condition to ensure that rq is indeed tangent at point q. This approach will yield the desired value of x.