Final answer:
To find the equation of the tangent to the circle at point (20,21), calculate the negative reciprocal of the slope of the radius to the point, which is the slope of the tangent, then apply the point-slope form of a line to get the tangent's equation: y = (-20/21)x + 841/21.
Step-by-step explanation:
To calculate the equation of the tangent at the point (20,21) on a circle with center (0,0) and radius 29, we first need to understand that a radius of a circle is perpendicular to the tangent at the point of contact. Therefore, if we find the slope of the radius (which is a line from the center to the point (20,21)), the slope of the tangent line will be the negative reciprocal of this slope.
The slope of the radius is given by the rise over run, that is, (21 - 0) / (20 - 0) which simplifies to 21/20. The negative reciprocal of this slope will be -20/21. Now that we have the slope of the tangent, we can use the point-slope form of a line to find its equation. The point-slope formula is given by (y - y1) = m(x - x1) where m is the slope and (x1, y1) is the point on the line. Substituting our known values, we get:
(y - 21) = (-20/21)(x - 20)
This equation can be rewritten in slope-intercept form, which looks like y = mx + b. Multiplying both sides of our tangent equation by 21 to eliminate the fraction, we get:
21(y - 21) = -20(x - 20)
21y - 441 = -20x + 400
Therefore, the equation of the tangent in slope-intercept form is:
y = (-20/21)x + 841/21