Question

Can any conic have its focus lying on the corresponding directrix? Give reasons for
your answer.

Answer 1

Answer on Question #78892 – Math – Analytic Geometry

Question

Can any conic have its focus lying on the corresponding directrix? Give reasons for your answer.

Solution

You can define conic section as locus of a point which moves so that the ratio of its distance from a given point called focus and a given line called directrix is always constant.

In case of ellipse, this ratio $< 1$, in case of hyperbola, this ratio $> 1$ and in case of parabola, this ratio $= 1$.

What happens when focus is on the directrix itself? Consider the following diagram, where $F$ is focus and directrix is line $l_{1}$ so that $F$ lies on it. Consider another line $l_{2}$ passing through $F$, so that it forms an angle of $\theta$ with $l_{1}$.

Now consider any other point on $l_{2}$. Observe that the ratio of the distance of the point from focus $F$ and directrix $l_{1}$ will always be $\frac{1}{\sin \theta}$. In fact, there could be two lines as distance is scalar and independent of sign and two lines with angles $\theta$ as well as $(180{}^{\circ} - \theta)$ will both have the same ratio.

Hence such a conic section will be a pair of lines.

**Answer**: Yes, such a conic section will be a pair of lines.

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