Question

How many other conditions can a conic section satisfy when we are given (1) its centre, (2) its focus, (3) its eccentricity, (4) its axes, (5) a tangent, (6) a tangent and its point of contact, (7) the position of one of its asymptotes?

Answer 1

$\left(i\right)$ Taking the origin at center, the equation of any conic may be written as $a{x}^2+2hxy+b{y}^2=1$.
So this equation has $3$ unknowns a,h and b. Hence $3$ conditions are required.
$\left(ii\right)$ Taking the origin at the focus, the equation of any conic may be given as $x^2+y^2=e^2{(x\cos \alpha +y\sin \alpha -p)}^2$
We have $3$ constants, i.e., e, $\alpha$ an p
So, $3$ conditions are required.
$\left(iii\right)$ Taking the origin at the point $(h,k)$ the equation of the conic may be written as
${(x-h)}^2+{(y-k)}^2=e^2{(x\cos \alpha +y\sin \alpha -p)}^2$
Here, e being give, the other unknowns are h,k, $\alpha$ and p. So, $4$ conditions are required.
$\left(iv\right)$ If the position of axes be given, the equation of conic may be given by $a{x}^2+2hxy+b{y}^2=1$, where a and b constants. Hence to obtain them, $2$ conditions are required.
$\left(v\right)$A conic may touch any five lines, one line being given, so other $4$ are to be known, so $4$ more conditions are required.
$\left(vi\right)$ As we want $5$ conditions to determine a conic is here $2$ being given
(a) it is tangent and (b) the point of contact only $3$ more conditions are required.
$\left(vii\right)$ An asymptote being nothing but a tangent at a given point (i.e., at infinity), so as in $\left(iv\right)$ only $3$ more conditions are required.