Tangent lines for a hyperbola Find an equation of the line tan...

Question

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

Accepted Answer

Welcome back, everyone. Find the equation of the tangent line to the hyperbola X2 divided by 9 minus Y2 divided by 4 equals 1 at the 0.6.2 square root of 2. Write your answer in slope intercept form. For this problem, let's remember the tangent line equation for the hyperbola. Specifically, it would be x0 multiplied by x divided by a quad minus Y0 multiplied by Y divided by B2 equals 1. What we want to do is simply understand that X 0 Y 0. These represent the coordinates of the point of tangency. According to the context of the problem, this is our 0.6.2 square root of 2. Now, a squad is basically the denominator of X2 divided by 9, right? So A2 is equal to 9 and B2 is the denominator of negative Y2 divided by 4. So B2 is equal to 4. And now we're going to substitute this into the equation of the tangent line. So x0 is 6. We multiply that by X. We divide by a squad, which is 9. We subtract Y0 multiplied by Y, so we subtract 2 square root of 2 multiplied by Y divided by B2, which is 4, and this is equal to 1. And that's it. What we want to do is simply rearrange this equation and solve for a Y. So now, what we can do is simply find the common denominator of 9 and 4, that will be 9 multiplied by 4 and that's 36. So we can simply rewrite. Everything under one fraction. Our common denominator is 36, so the first term becomes 6 x multiplied by 4. That's 24 X, and we're subtracting 2 square roots of 2 multiplied by Y multiplied by 9. So we get minus 18 square root of 2 Y. And on the right hand side we have one. From here, we can multiply both sides by 36, right? And we get 24 X minus 18 square root of 2 Y equals 36. From here, our goal is to solve for Y, so what we can do is simply move. Our term negative 18 square root of 2 to the right. Right? Because it has a negative sign on the left. We want to ensure it's positive, and we're going to move 36 to the left. So we get 24 X minus 36 equals 18 square root of 2Y. And then Y is going to be equal to, we can divide both sides by 18 square root of 2. So we get 24 divided by 18 square root of 2. Multiplied by X minus 36 divided by 18 square root of 2. Well done. And now what we want to do is simplify. We can show that Y is equal to now 24 divided by 18. We can simplify and we can divide both sides by 6. So we get 4 divided by. 3 square root of 2 multiplied by X minus 36 divided by 18 is 2. So we're subtracting 2 divided by square root of 2. And from here we can just rationalize each denominator. So Y is going to be equal to or square root of 2 divided by we multiplied by 2 gives us 6, right? Minus 2 divided by square root of 2, gives us 2 square roots of 2 divided by 2 or basically, we're just subtracting square root of 2, right? And L 4 divided by 6, we can simplify this expression as well. We can just divide both sides by 2. So we get 2 square roots of 2. Divided by we. For the fractional term. And our equation becomes Y equals 2 square root of 2 divided by 3 multiplied by X minus square root of 2, which is our final answer for this problem. Thank you for watching.

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

Key Concepts

Implicit Differentiation

Equation of a Tangent Line

Hyperbola Equation and Properties

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