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I must find the two tangents that pass through the point $(2,7)$ for the ellipse $2x^2+y^2+2x-3y-2=0$ all I was able to was getting $\frac{dy}{dx}=\frac{(-4x-2)}{(2y-3)}$ and therefore equalizing that with the point I'm given $\frac{(-4x-2)}{(2y-3)}=\frac{7-y}{2-x}$ so... please help me to find these two tangets

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  • $\begingroup$ Do you know how to find tangents to an ellipse in standard position? This ellipse is just a translation of one. $\endgroup$ – amd Dec 10 '18 at 7:58
  • $\begingroup$ Yes I know how to do it if it's centered , or at least I have done successfully if it's centered at (0,0) $\endgroup$ – rorod8 Dec 10 '18 at 16:49
  • $\begingroup$ OK, so translate the ellipse so that it’s centered, then translate the two lines back to the original position. $\endgroup$ – amd Dec 10 '18 at 20:40
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Hint:

The equation of any straight line passing through $(2,7)$ can be set as $$\dfrac{y-7}{x-2}=m$$ where $m$ is the gradient.

Replace the value of $y$ in the equation of the ellipse to form a quadratic equation in $y$

whose two roots represent the ordinate of the intersection

For tangency, the two root must be same.

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Since you know how to find the tangents to an ellipse that’s centered on the origin, a way to approach this problem is to translate this ellipse so that its center is at the origin (and of course translate $(2,7)$ by the same amount), find the tangents, and then translate them back.

The ellipse’s center can by found in various ways such as computing partial derivatives and setting them equal to zero: the center of the ellipse is the solution to the equations $$\begin{align} 4x+2 &= 0 \\ 2y-3 &= 0. \end{align}$$ Translate the origin to the center point $(x_c,y_c)$ by making the substitutions $x\to x+x_c$ and $y\to y+y_c$ in its equation. A short cut: doing this will leave the quadratic terms alone, eliminate the first-degree terms in $x$ and $y$ and replace the constant term with the value of the left-hand side of the original equation at $(x_c,y_c)$, i.e., the translated equation will be $$2x^2+y^2+(2x_c^2+y_c^2+2x_c-3y_c-2)=0.$$ Find the tangents to this ellipse that go through the point $(2-x_c,y-y_c)$. Once you’ve done this, translate the equations of the lines back to the original coordinate system with the substitutions $x\to x-x_c$ and $y\to y-y_c$.

A more direct approach is to use the polar of the point $(2,7)$, which is a line that includes the chord of contact of the tangents through the point. The equation of the polar of a point $(x_0,y_0)$ can be found mechanically by making the following substitutions in the equation of the conic: $$x^2 \to xx_0 \\ y^2 \to yy_0 \\ xy \to \frac12(xy_0+x_0y) \\ x \to \frac12(x+x_0) \\ y \to \frac12(y+y_0).$$ Find the intersections of the resulting line with the ellipse and derive the equations of the lines through these points and $(2,7)$.

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