Intereting Posts

$-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$
Why is every positive integer the sum of 3 triangular numbers?
The group of invertible linear operators on a Banach space
Finding the largest subset of factors of a number whose product is the number itself
Total Time a ball bounces for from a height of 8 feet and rebounds to a height 5/8
A path to truly understanding probability and statistics
If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$
prove that $gcd(f_m, f_n) = f_{gcd(n, m)}$, where $f_n$ is the nth Fibonacci number.
Importance of determining whether a number is squarefree, using geometry
Are the smooth functions dense in either $\mathcal L_2$ or $\mathcal L_1$?
Why isn't the derivative of a rotation matrix skew symmetric?
Number of spanning trees in a ladder graph
Could someone show me a simple example of something being proved unprovable?
Sum of fourth powers in terms of sum of squares
Some questions about the gamma function

Suppose that there is a hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$.

I would like to figure out an equation that describes tangent line to this hyperbola.

How would I be able to do this using calculus? My calculus trials are bring me some gibberish answers.

- Inequality for incomplete Gamma Function
- What concepts were most difficult for you to understand in Calculus?
- If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$
- Inequality $\sum\limits_{1\le k\le n}\frac{\sin kx}{k}\ge 0$ (Fejer-Jackson)
- Why is $\lim_{x \to c}g(f(x)) = g(\lim_{x \to c}f(x))$
- Convergence of $\sum_{n=1}^{\infty} \log\left(\frac{(2n)^2}{(2n+1)(2n-1)}\right)$

I did use the calculus below, but what I want is how to derive the general form equation of $y=mx \pm \sqrt{a^2m^2-b^2}$ where $m$ refers to the slope of a tangent line.

- Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
- Compute polylog of order $3$ at $\frac{1}{2}$
- Which universities teach true infinitesimal calculus?
- Show that if $f(x)=\int_{0}^x f(t)dt$, then $f=0$
- Some integral with sine
- Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?
- How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion?
- Geometric derivation of the quadratic equation
- Crafty solutions to the following limit
- Show $\lim\limits_{n\to\infty} \frac{2n^2-3}{3n^ 2+2n-1}=\frac23$ Using Formal Definition of Limit

(1a)

Let $y=mx+c$ be a tangent of $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

Let $(h,k)$ be the point of intersection.

So, $k=mh+c, b^2h^2-a^2k^2=a^2b^2\implies b^2h^2-a^2(mh+c)^2=a^2b^2$

or,$h^2(b^2-a^2m^2)- 2a^2mch -a^2b^2-a^2c^2=0 $

This is a quadratic equation in $h,$ for tangency, the roots need to be same, to make the two points of intersection coincident.

$\implies (- 2a^2mc)^2=4\cdot (b^2-a^2m^2)(-a^2b^2-a^2c^2) $

$a^2m^2c^2=a^2m^2b^2+a^2m^2c^2-b^4-b^2c^2$ cancelling out $a^2$ as $a\ne0$

$0=a^2m^2b^2-b^4-b^2c^2$

$0=a^2m^2-b^2-c^2$ cancelling out $b^2$ as $b\ne0$

$\implies c^2=a^2m^2-b^2\implies c=\pm\sqrt{a^2m^2-b^2}$

(1b)

Alternatively, we can take the central conic to be $Ax^2+By^2=1$

Applying the same method we get, $ABc^2=A+m^2B$

Here $A=\frac 1{a^2},B=-\frac 1{b^2}\implies c^2=a^2m^2-b^2$

(2a)

Using calculus as André Nicolas has already done,

we find the equation of the tangent to be $$\frac {xx_1}{a^2}-\frac{yy_1}{b^2}-1=0$$

Comparing with $mx-y+c=0$ we get, $$\frac{x_1}{ma^2}=\frac{y_1}{b^2}=\frac{-1} c$$

Now $(x_1,y_1)$ lies on the given curve, so $\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}=1$, eliminating $x_1,y_1$, we shall get the desired result.

(2b) we know , the parametric equation of the given curve is $x=a\sec t,y=b\tan t$

So, the equation of the tangent becomes $\frac{x\sec t}a-\frac{y\tan t}b-1=0$

Comparing with $mx-y+c=0$ we get, $$\frac{\sec t}{ma}=\frac{\tan t}b=\frac {-1}c$$

So, $\sec t=-\frac{ma}c,\tan t=-\frac b c$

Now use eliminate $t$.

It is such a nice equation that we might as well differentiate with respect to $x$ **immediately**. We get

$$\frac{2x}{a^2}-\frac{2yy’}{b^2}=0.$$

Solve for $y’$.

**Remark:** Alternately, we can solve for $y$ in terms of $x$. We get *two* equations. Then differentiate. A bit messier than the differentiation above, but doable.

The device we used, *implicit differentiation*, is very useful. In or situation, we can think of the differentiation of $y^2$ with respect to $x$ as using the Product Rule, or the Chain rule. Note that $y^2$ is a product. Its derivative with respect to $x$ is $yy’+y’y=2yy’$.

- Representability as a Sum of Three Positive Squares or Non-negative Triangular Numbers
- Localization and direct limit
- Logarithm of a Markov Matrix
- $k$-element subsets of $$ that do not contain $2$ consecutive integers
- Good textbooks on homological algebra
- Range conditions on a linear operator
- Closure of a number field with respect to roots of a cubic
- proving that $\frac{(n^2)!}{(n!)^n}$ is an integer
- Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$
- Geometric interpretation of complex path integral
- Condition for 3 complex numbers to represent an equilateral triangle
- Topology on $R((t))$, why is it always the same?
- Fermat numbers are coprime
- Locally compact nonarchimedian fields
- Induction on Real Numbers