Final answer:
The equations of the tangent and normal to the ellipse 4x²+9y²=20 at the point (1,4/3) are derived by first finding the derivative of the ellipse's equation to obtain the slope, and then applying the point-slope form.
Step-by-step explanation:
To find the equations of the tangent and normal to the given ellipse 4x²+9y²=20 at the point (1,4/3), we first need to find the derivative of the ellipse's equation to get the slope of the tangent line at the given point.
We implicitly differentiate the equation of the ellipse with respect to x:
8x + 18yy' = 0
Now solve for y':
y' = -⅓x/y
Plugging in the point (1,4/3), we get the slope of the tangent:
y' = -⅓(1)/(4/3) = -⅓/4
Using the point-slope form, the equation of the tangent line is:
y - ⅔/3 = -⅓/4(x - 1)
For the normal line, it will have a slope that is the negative reciprocal of the tangent's slope:
The slope of the normal line is 4/3. So, the equation of the normal line using the point-slope form is:
y - ⅔/3 = 4/3(x - 1)
Thus, we've derived the equations for both the tangent and the normal at the given point on the ellipse.