Results 1 to 7 of 7

Thread: Tangent Passing Through a Point & Implicit Differentiation

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    78

    Tangent Passing Through a Point & Implicit Differentiation

    Question:
    Find the equation(s) of the tangents to $\displaystyle x^2 + 4y^2 = 16$ which pass through
    a) the point (2,2)
    b) the point (4,6).

    Attempted Solution:
    $\displaystyle y = \pm\sqrt{\frac{-x^2 + 16}{4}}$

    $\displaystyle y' = \frac{-x}{4y}$

    a) For $\displaystyle P(2,2)$:

    Let $\displaystyle P_{tan}$ be parametrized as $\displaystyle (a, f(a)$.

    $\displaystyle \frac{2 - \pm\sqrt{\frac{-a^2 + 16}{4}}}{2 - a} = \frac{rise}{run}= m$

    What do I do from here? The points given are not points of tangency; they are points that the tangents pass through. So, do I have to set something equal and solve for $\displaystyle a$?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    $\displaystyle \frac{y-2}{x-2}=[\frac{dy}{dx}]_{(x,y)=(2,2)}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2010
    Posts
    78
    Do I have to sub P(2,2) into the derivative? What exactly am I trying to solve for?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    after gettin $\displaystyle \frac{dy}{dx}$, substitute x by 2 and y by 2.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Mar 2010
    Posts
    78
    That leaves me with a slope of 1/4. Do I set 1/4 equal to the rise/run equation in my OP?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    That leaves you with the equation $\displaystyle \frac{y-2}{x-2}=1/4$
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Sambit View Post
    $\displaystyle \frac{y-2}{x-2}=[\frac{dy}{dx}]_{(x,y)=(2,2)}$
    We need to take care as the point $\displaystyle (2,2)$ is not on the ellipse.

    If you plot the question both answers can be read off the graph. The tangent lines must be horizontal and vertical. Note that

    $\displaystyle \displaystyle x^2+4y^2=16 \iff \frac{x^2}{4^2}+\frac{y^2}{2^2}=1$

    Is the equation in standard form.

    Tangent Passing Through a Point & Implicit Differentiation-ellipse.jpg



    To solve it algebraically we need to find a point $\displaystyle (a,b)$ on the ellipse e.g it must satisfy $\displaystyle a^2+4b^2=16$ and

    $\displaystyle \frac{b-2}{a-2}=\frac{dy}{dx}\bigg|(a,b)=-\frac{a}{4b}$

    Multiplying out the bottom equation gives

    $\displaystyle 4b^2+a^2=2a+8b$ but since it must be on the ellipse $\displaystyle a^2+4b^2=16$ this gives

    $\displaystyle 16=2a+8b \implies a=8-4b=4(2-b)$

    Now putting this back into the ellipse again gives

    $\displaystyle [4(2-b)]^2+4b^2=16 \iff 5b^2-16b+12=0 \iff (5b-6)(b-2)=0$ as you can check this gives $\displaystyle b=2 \implies a=0$ and correctly gives the point $\displaystyle (0,2)$ as when one of the tangent lines occur.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Feb 28th 2012, 03:27 PM
  2. Horizontal Tangent - Implicit Differentiation
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Dec 7th 2010, 02:01 PM
  3. implicit differentiation, tangent line help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 17th 2010, 02:48 PM
  4. implicit differentiation to tangent line
    Posted in the Calculus Forum
    Replies: 15
    Last Post: Nov 5th 2009, 10:32 AM
  5. Implicit differentiation and tangent lines
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Oct 8th 2009, 03:38 PM

Search Tags


/mathhelpforum @mathhelpforum