Final answer:
To find the equation of the tangent to the curve at the given point, we need to find the derivative of the curve at that point and then use the slope and the point to form the equation of the tangent line.
Step-by-step explanation:
To find the equation of the tangent to the curve at the given point (6,9), we need to first find the derivative of the curve at that point. Let's start by finding the derivative of x with respect to t: dx/dt = 2t - 1. To find the derivative of y with respect to t, dy/dt, we differentiate y = t^2 + 8 with respect to t, giving us dy/dt = 2t.
The slope of the tangent line is equal to the slope of the curve at the given point (6,9), which is dy/dt at t = 6. Plugging in t = 6 into dy/dt, we get the slope of the tangent line. Then we can use the equation of a line y = mx + c, where m is the slope and (x,y) is a point on the line, to find the equation of the tangent line.
So, slope of tangent line = dy/dt at t = 6 = 2(6) = 12. The equation of the tangent line is y - 9 = 12(x - 6), or y = 12x - 63.