What value of x makes line rq tangent to circle p at point q?

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To find the value of x that makes line rq tangent to circle p at point q, we first need to understand the geometric relationship between the line and the circle.

A tangent line to a circle intersects the circle at exactly one point. In this scenario, point q is the point of tangency. The key property of tangents is that the radius drawn to the point of tangency is perpendicular to the tangent line.

Let’s assume that circle p has a center at coordinates (h, k) and a radius of r. The equation of the circle can then be described as:

(x - h)² + (y - k)² = r²

Suppose point q has coordinates (x_q, y_q) which lies on the circle. For rq to be tangent at point q, the following conditions must be met:

  1. The distance from the center of the circle, point p, to point q must equal the radius, r:
    2. sqrt((x_q - h)² + (y_q - k)²) = r
  2. The slope of line pq (from the center of the circle p to the point of tangency q) must be calculated. If the coordinates of point r are (x_r, y_r), then the slope of line pq is:
    3. m_{pq} = (y_q - k) / (x_q - h)
  3. The slope of line rq must be the negative reciprocal of the radius slope to ensure perpendicularity:
    4. m_{rq} = -1/m_{pq}

By setting up the system of equations and taking into account the geometry involved, one can solve for x given a specific set of coordinates for r and the circle’s parameters.

To summarize, you need to formulate the equations for the circle, find the coordinates of point q, determine the slopes, and solve for x based on the provided parameters. This approach will yield the particular value of x that ensures rq is tangent to circle p at point q.

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