Find the equation of the tangent line to the ellipse at the given point

Question

(x²/a²) + (y²/b²)=1   Point (m,n)  

 

Answer should be in terms of x,y,m,n,a,b and answer is _______=1

Answer 1

The slope of the ellipse at the point (m,n) can be computed by implicit differentiation of the ellipse equation with respect to x.        Implicit differentiation yields:

 

2x/a² + (2 y/b²) (dy/dx) = 0    The slope is dy/dx.    This can be rearranged as:

 

slope =   - (x/y) b²/a²     At the point (m, n) the slope will be

 

  slope = - (m/n) b²/a²

 

The equation for the tangent line can be found using the formula for a line when the slope and one point are known.

 

(y - n) = slope ( x - m)    =   -(m/n) (b²/a²) ( x - m)

 

After a lot of algebra, this can be reorganized into the form:

 

 a² n y/(a²n² + b² m²)  + b² m x /(a² n² + b² m²)  =  1

 

This is the equation of a straight line in standard form.

 

Answer 2

To find the equation of the line, you need the slope and a point. You have a point (m, n), so you need the slope m = dy/dx.

 

d/dx(x²/a² + y²/b² = 1)

 

Take the derivative on x normally, but with y, you need to remember the chain rule.

2x/a² + 2y/b²*dy/dx = 0

 

Now you can solve this for dy/dx to find the general slope of this curve. To find the slope at point (m, n) plug m and n in place of x and y.

 

Finally, use point slope form of a line, and you're done.

y-n = dy/dx (x-m)