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Prove: The line tangent to the hyperbola$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$at the point $\left(x_{0}, y_{0}\right)$ has the equation$$\frac{x x_{0}}{a^{2}}-\frac{y y_{0}}{b^{2}}=1$$

$\frac{\pi_{0} x}{a^{2}}-\frac{y_{0} y}{b^{2}}=1$

Calculus 1 / AB

Calculus 2 / BC

Chapter 10

PARAMETRIC AND POLAR CURVES; CONIC SECTIONS

Section 4

Conic Sections

Derivatives

Differentiation

Parametric Equations

Polar Coordinates

Applications of the Derivative

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discipline or 50. We need to prove the line attention to the hyper bola given hyper valet's X squared by a squared minus y squared by B squared equal to one at ex not from a Why not? Is x x not by a square minus Why? Why not buy base quite equal to one? So we have to prove this. Let us suppose this to be question number one. Now we will be starting with, uh, this point. It's not violent. Why not lies on the parabola? So the question number one becomes it's not a square by square, minus whiner square. A B is quite equal toe one. So if you need to find the value of violent square by B squared, this will be why not a square. But be square, excellent squired by a square minus one. So we will be using this value afterwards. Now let us try and Royal off a sketch off the hyper Gola and the tangent. Okay, I probably will look like this. And if this is any point x, not comma, why not? So tangent would look like Yes, And when that's not why it is the point of tendency So first of all, we'll be finding the slope of pendant slope off the tangent. Okay, stop off any present at this hyper Bhola, which will be found out by differentiating the generally question. All given a question off Hyper bola with respect to X minus y squared by B squared equal toe D by D. X one. So since differentiation follows distributive law so this will be DVD X axis Squire by a Squire minus DVD X. Why squired by B squared equal to zero because constant term have a differences in equal to zero No one by a square and one baby square are constants. So this can be taken outside the i d X access square minus one by B squared DVD X Waske or equal to zero. So access choir has differentiation to works. So two x by is choir minus to Why by be square Do you ever the X equal to zero? This condition is to why, by being squired de vie ready X equal took two works by a square nor multiply both sides by being square by two y This will become Do you have any X equal to to expire a Squire and to be square by two. Why do you will get canceled out? So be squired my square. That's my wife. So this is the slow attendant at any general point. For in a particular point we confined slow. We can find the slow attendant at It's not common. Why not? So we'll be plugging the value off X as X not. And whereas we're not, this will be be a square X not by is square. Why not? Now we're building using this value to write the question of tangent. So we appoint next month. Why? Why not? We have slope so point slow form off a straight land. Why minus why not equal toe d y by d. X That slope at It's not Why not x minus x Not so. Why am I? So I am not Will be equal toe Be a Squire X not my Esquire. Why not X minus X? Not okay. So let us cause Multiply with Why not? Why minus why not into Why not equal toe b squared ex not X minus that's not multiply. Both multiply or opening the back it with Why not his in distributive law? Why Why not? Minus one. The square equal Toby Squired ex Not this is square with luminess it is X minus X not not divide. Both sides were b squared. It will be Why? Why not by the square minus Why not a square by B squared equal toe X marked by a square X minus X Not now. We have already found the value of wise. Why not square the B squared? Which waas here? X marks Squire by a square minus one. So let us plug in. Okay, this is why? Why not? My b squared minus X More squared. Bye. A square minus one. Now it is open the bracket XX not by a square minus x more squired by a square. So why? Why not to square the be square minus x not square by square Bless when equal toe xx not Why is choir minus x not square by a square. So this can begin sellout and little subtract why we're not based world from both the sides. We will be having xx marred by a square minus Why? Why not buy base square? So this can also be written as xx not my a square minus. Why? Why not by B squared equal to one, which is the required A question off the attendant. Thank you.

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