Final answer:
The equations of the tangent lines to the circle at the points where x = 4 are y - 3 = -4/3(x - 4) and y + 3 = 4/3(x - 4). The equations of the normal lines are y - 3 = 3/4(x - 4) and y + 3 = -3/4(x - 4).
Step-by-step explanation:
Finding Tangent and Normal Lines to a Circle
To find the equations of the tangent lines to the circle x^2 + y^2 = 25 at the points where x = 4, we first need to determine the y-coordinates of these points on the circle. Since the equation of the circle is x^2 + y^2 = 25, plugging in x = 4 gives us 4^2 + y^2 = 25, which simplifies to y^2 = 9. Thus, the y-coordinates are y = 3 and y = -3.
The gradient (slope) of the radius at these points is the opposite reciprocal of the gradient of the tangent. For a circle centered at the origin, the radius at the point (4, 3) has a slope of 3/4, so the slope of the tangent line must be -4/3. Similarly, for the point (4, -3), the slope of the radius is -3/4 and that of the tangent line is 4/3.
To find the equation of the tangent lines, we use the point-slope form of a line, which is y - y1 = m(x - x1). Inserting the respective points and slopes gives us the equations of the tangent lines: y - 3 = -4/3(x - 4) and y + 3 = 4/3(x - 4).
For the normal lines, which are perpendicular to the tangent lines at the points of tangency, we use the negative reciprocal of the tangent slopes. Thus, the slopes of the normal lines are 3/4 and -3/4, respectively. The equations of the normal lines are y - 3 = 3/4(x - 4) and y + 3 = -3/4(x - 4).