Finding the Tangent to an Ellipse (using calculus)

[goingforward]

Hi
i was given this question for homework:

An ellipse has the equation x^2+5y^2=5
a line has the equation y=mx+c

show that if the line is a tangent to the ellipse then c^2=5m^2+1

I understand there are two ways to do this (fermat's method and calculus) but i need to do it the calculus way.

so the derivative of x^2+5y^2=5 was dy/dx=-x/5y and at the point (p,q) m=-p/5q
y-y1=m(x-x1)
y-q=-p/5q(x-p)
y=-px/5q + p^2/5q + q
y=(-p/5q)x + (p^2/5q + q)
so then c=p^2/5q + q
the problem is (p^2/5q + q)^2 doesnt equal 5(-p/5q)^2+1

Help??

 

[Deleted member 4993]

Hi
i was given this question for homework:

An ellipse has the equation x^2+5y^2=5
a line has the equation y=mx+c

show that if the line is a tangent to the ellipse then c^2=5m^2+1

I understand there are two ways to do this (fermat's method and calculus) but i need to do it the calculus way.

so the derivative of x^2+5y^2=5 was dy/dx=-x/5y and at the point (p,q) m=-p/5q
y-y1=m(x-x1)
y-q=-p/5q(x-p)
y=-px/5q + p^2/5q + q
y=(-p/5q)x + (p^2/5q + q)
so then c=p^2/5q + q
the problem is (p^2/5q + q)^2 doesnt equal 5(-p/5q)^2+1

Help??

How about using the fact that:

p^2 + 5q^2 = 5

 

[goingforward]

Thanks! I get it now!

 

[lookagain]

Hi
i was given this question for homework:

An ellipse has the equation x^2 + 5y^2 = 5
a line has the equation y = mx + c

show that if the line is a tangent to the ellipse then c^2 = 5m^2 + 1

I understand there are two ways to do this (fermat's method and calculus) but i need to do it the calculus way.

so the derivative of x^2 + 5y^2 = 5 was dy/dx = -x/(5y) and at the point (p,q) m = -p/(5q)

y - y1 = m(x - x1)

y - q = [-p/(5q)](x - p)

y = -px/(5q) + p^2/(5q) + q

y = [-p/(5q)]x + [p^2/(5q) + q]

so then c = p^2/(5q) + q

the problem is [p^2/(5q) + q]^2 doesnt equal 5[-p/(5q)^2] + 1

Help??

goingforward,

if you want to go forward, then make sure you use needed grouping
symbols as suggested above in the amended quote box. Also, you
should add vertical and horizontal spaces for better readability.