Ellipse and locus of point of intersection of two tangents

• April 19th 2010, 05:04 AM
arze
Ellipse and locus of point of intersection of two tangents
Prove that the line lx+my+n=0 is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=0$ if $a^2l^2+b^2m^2=n^2$.
Tangents are drawn from a point P to the ellipse. Lines from the origin are drawn perpendicular to the tangents. If these lines are conjugate diameters, prove that the equation of the locus of P is $a^2x^2+b^2y^2=a^4+b^4$

I have proved the first part.
For this next part the equation of a tangent to an ellipse in the form
$y=mx\pm\sqrt{a^2m^2+b^2}$
$\frac{1}{m_1}.\frac{1}{m_2}=-\frac{b^2}{a^2}$
$m_2=-\frac{a^2}{b^2{m_1}^2}$

I used the two gradients, m, in two equations for tangents and tried to solve for x and y, am I doing it right? because I get very very long and difficult expressions.
Thanks!
• April 19th 2010, 01:35 PM
Opalg
Quote:

Originally Posted by arze
Prove that the line lx+my+n=0 is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=0$ if $a^2l^2+b^2m^2=n^2$.
Tangents are drawn from a point P to the ellipse. Lines from the origin are drawn perpendicular to the tangents. If these lines are conjugate diameters, prove that the equation of the locus of P is $a^2x^2+b^2y^2=a^4+b^4$

I have proved the first part.
For this next part the equation of a tangent to an ellipse in the form
$y=mx\pm\sqrt{a^2m^2+b^2}$
$\frac{1}{m_1}.\frac{1}{m_2}=-\frac{b^2}{a^2}$
$m_2=-\frac{a^2}{b^2{m_1}^2}$

For the last part, let P be the point $(p,q)$. The tangent $lx+my+n=0$ satisfies $a^2l^2+b^2m^2=n^2$, and it passes through P if $ap+bq=n$. Substitute n from the second of those two equations into the first equation: $a^2l^2+b^2m^2 = (lp+mq)^2$, which can be written as

$(a^2-p^2)\bigl(\tfrac lm\bigr)^2 -2pq\tfrac lm + (b^2-q^2) = 0.\qquad(*)$

The two roots of the quadratic equation (*) in $\tfrac lm$ are $\tfrac lm = -m_1$ and $\tfrac lm = -m_2$, where $m_1,\,m_2$ are the gradients of the tangents through P (because $-\tfrac lm$ is the gradient of the line $lx+my+n=0$). Therefore $m_1m_2 = \frac{b^2-q^2}{a^2-p^2}$ (formula for product of roots of quadratic: constant term divided by coefficient of $x^2$). But you already know that $\frac{1}{m_1}\mathord{\cdot}\frac{1}{m_2}=-\frac{b^2}{a^2}$. Put those two expressions for $m_1m_2$ together to see that $\frac{b^2-q^2}{a^2-p^2} = -\frac{a^2}{b^2}$, from which $a^2p^2 + b^2q^2 = a^4+b^4$. Finally, replace $(p,q)$ by $(x,y)$ to get the locus of P.