Why does the intercept form of the straight line equation $x/a+y/b=1$ look similiar to the standard equation of an ellipse $x^2/a^2 +y^2/b^2 =1$

Question

Today we started 2-D geometry in school covering straight lines and conic sections. And I just had this shower thought, why do these two standard equations look similiar. Is there any implication of any concept, I am failing to recognize here?

Answer 1

There's a lot to be said on that subject, but it might be early to get into it.

Note if you use $|x/a|+|y/b|=1$ instead you get an inscribed quadrilateral within the ellipse.

Graph both in desmos and play around with $a$ and $b$ to see what happens. You'll find changing them stretches and squeezes both figures. That's because $a$ and $b$ play the role of scaling factors, sort of magnifiying or shrinking one dimension of the graph.

The most general form of a conic sectoin is $Ax^2+By^2+Cxy+Dx+Ey+F=0$ with some algebra find those six variables imply some geometrically more relevant information. THey imply the before mentioned scaling factors, $C$ has a lot to say about whether the conic's axes of symmetry or parallel to the coordinate axes. They also tell you how far away the vertices and focal points are from the origin.

$(x/a)^n+(y/a)^n=1$ stops being a conic for $n>2$, but has some interesting features, especially regarding metric spaces. Roughly speaking $n=1$ implies a straight line distance, $n=2$ implies the Pythagorean Theorem and typical distance notions of euclidean geometry, $n=\infty$ actually becomes a veration of the Max function $max(x,y)=x \iff x\ge y$ .

Answer 2

In coordinate geometry, it's common approach to rearrange an equation wisely in more trivial or intuitive manner.

• Intercept form of a straight line

$$\frac{x}{a}+\frac{y}{b}=1 \tag{$L$}$$

• Standard ellipse based on semi-axes

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \tag{$E$}$$

For the two mentioned cases, the $x$-intercept $A(a,0)$ or $y$-intercept $B(0,b)$ is automatically satisfying both sides of an equation.

Moreover, their orders of contact will be zero-th and first for $L$ and $E$ respectively.

See further examples usually appears in analytical geometry below:

• Polar of $E$ w.r.t. a pole $P(x',y')$

$$\frac{x'x}{a^2}+\frac{y'y}{b^2}=1$$

• Tangent pair of $E$ from $P$

$$ \left( \frac{x'x}{a^2}+\frac{y'y}{b^2}-1 \right)^2 = \left( \frac{x'^2}{a^2}+\frac{y'^2}{b^2}-1 \right) \left( \frac{x^2}{a^2}+\frac{y^2}{b^2}-1 \right)$$

• Family of conics with given $x$ and $y$ intercepts

$$\frac{(x-\alpha)(x-\beta)}{\alpha \beta}+\frac{(y-\gamma)(y-\delta)}{\gamma \delta}+2hxy=1$$

• Family of conics touching the axes at $A$ and $B$

$$ \left( \frac{x}{a}+\frac{y}{b}-1 \right)^2 =\frac{2\lambda xy}{ab} \tag{$\lambda \ne 0$}$$

• An oblique conic touches a standard ellipse at $(a\cos \theta,b\sin \theta)$

$$ \frac{x^2}{a^2}+ \frac{y^2}{b^2}-1=k \left( \frac{x\sin \theta}{a}- \frac{y\cos \theta}{b} \right)^2$$

• Family of conics touching $\Delta OAB$, both internally and externally

$$ \left( \frac{x}{\lambda a}+\frac{y}{\mu b}-1 \right)^2=\frac{4xy}{ab} \left( \frac{1}{\lambda}-1 \right) \left( \frac{1}{\mu}-1 \right)$$

• See further application of pole-and-polar relation in $3$-D case here.