This is the question,
Equation of the tangent at the point (3, -1) to the ellipse $2x^2 + 9y^2 = 3$ is
Solution
my question is, shouldn't the tangent pass through the point given?
one of the option was $2x + 3y = 3$
And what if the question was like
Equation of the tangent from the point (3, -1) to the ellipse $2x^2 + 9y^2 = 3$ is
the only change is from the point can I still use this formula? and should the points satisfy the tangent in this case also?
Best Answer
It seems you've been exposed to some part of Joachimsthal's notation, in particular the $s_1=0,$ which is the tangent when the point is on the conic, but which more generally defines the polar of the point when the point is not on the conic.
There are more Joachimsthal notations. $s_1^2=s \cdot s_{11}$ is the one for your situation.
Let's factor $s_1^2-s \cdot s_{11},$ where the point is $(3,-1)$ and $s=0$ is your ellipse equation.
$$(2x^2+9y^2-3)(2(3)^2+9(-1)^2-3)-(2x\cdot 3+9y\cdot (-1)-3)^2)=\\3\cdot (3y+2x-3)(15y+2x+9)$$
As you can see, the two tangents fall right out.