Final answer:
The two real values of a that satisfy the equation (x-4) +(y-2) = 13 are a = 7 and a = 1.
The equation of the tangents drawn to the circle x² + y² - 2x+4y=0 from (0, 1) are y = -4x + 18 and y = -4x - 14.
Step-by-step explanation:
To find the values of a that satisfy the equation, we can substitute x = a and y = 0 into the equation of the circle.
This gives us the equation (a - 4)² + (0 - 2)² = 13. Simplifying this equation, we get (a - 4)² + 4 = 13.
Rearranging, we have (a - 4)² = 9. Taking the square root of both sides, we get a - 4 = ±3.
Adding 4 to both sides, we find that a = 7 or a = 1.
To find the equation of the tangents drawn to the given circle from the point (0, 1), we can first find the equation of the line passing through (0, 1) and the center of the circle.
The center of the circle is (4, 2) (from the equation (x-4) +(y-2) = 13).
The slope of the line passing through (0, 1) and (4, 2) can be found using the formula (y2 - y1) / (x2 - x1).
The slope is (2 - 1) / (4 - 0) = 1/4.
Using the equation y - y1 = m(x - x1), we substitute the values and simplify to find the equation of the line passing through (0, 1) and the center of the circle: y - 1 = (1/4)(x - 0).
Simplifying further, we get y = (1/4)x + 1.
The equation of the tangents drawn to the circle from (0, 1) will be the equations of the lines that are perpendicular to the line passing through (0, 1) and the center of the circle, and also pass through the points of tangency.
The equation of a line perpendicular to y = (1/4)x + 1 will have a slope that is the negative reciprocal of 1/4, which is -4.
Using the equation y - y1 = m(x - x1), we can use the slope-intercept form y = mx + b to find the equation of the perpendicular lines passing through the points of tangency.
For the first tangent, we substitute the values (x1, y1) = (4, 2) into the equation: y - 2 = -4(x - 4).
Simplifying, we get y = -4x + 18.
For the second tangent, we substitute the values (x1, y1) = (4, 2) into the equation: y - 2 = -4(x - 4).
Simplifying, we get y = -4x - 14.