# Equation of tangents of hyperbolas

• phospho
In summary, to find the equation of the tangents to the hyperbola H with equation \frac{x^2}{25} - \frac{y^2}{16} = 1 at the point (1,4), we need to use the relation a^2m^2 = b^2+c^2 and the equation of a tangent line y = mx+c. By differentiating the hyperbola with respect to x and evaluating at (1,4), we can find the value of m, the gradient of the tangent. However, the relation a^2m^2 = b^2+c^2 does not apply to tangent lines. Instead, the equation of the tangent line can be found by
phospho
find the equation of the tangents to the hyperbola H` with equation $\frac{x^2}{25} - \frac{y^2}{16} = 1$ at the point (1,4)

in an earlier part of the equation we had to prove that a tangent to the a hyperbola in the form of $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is in the form of $a^2m^2 = b^2 + c^2$ where the tangent is in the form of y = mx + c

so I differentiated with respect to x and got

$$\dfrac{dy}{dx} = \dfrac{16x}{25y}$$

subbing in the values of x and y I get the value of dy/dx to be 4/25

dy/dx is the gradient of the tangent so subbing that into the equation $a^2m^2 = b^2 + c^2$ as well as the values for a^2 and b^2 I can't get any real values for the constant, c so I'm not sure where I've gone wrong.

You have calculated m from evaluating dy/dx for the hyperbola at (1,4)

the equation of the tangent line is y = mx+c, so c (for the line) can be determined
by subbing in (1,4) for (x,y)

The relation a^2m^2 = b^2+c^2 is not that of a tangent line

SteamKing said:
You have calculated m from evaluating dy/dx for the hyperbola at (1,4)

the equation of the tangent line is y = mx+c, so c (for the line) can be determined
by subbing in (1,4) for (x,y)

The relation a^2m^2 = b^2+c^2 is not that of a tangent line

what is the relation of a^2m^2 = b^2+c^2 then?

SteamKing said:

understand now, thank you.

## 1. What is the equation of the tangent line to a hyperbola?

The equation of the tangent line to a hyperbola is given by y = mx + b, where m is the slope of the tangent line and b is the y-intercept.

## 2. How do you find the slope of the tangent line to a hyperbola?

The slope of the tangent line to a hyperbola can be found by taking the derivative of the hyperbola's equation and evaluating it at the point of tangency.

## 3. Can a hyperbola have more than one tangent line at a given point?

No, a hyperbola can have only one tangent line at a given point. This is because a tangent line must touch the hyperbola at only one point and have the same slope as the hyperbola at that point.

## 4. What is the relationship between the slopes of two intersecting tangent lines to a hyperbola?

The slopes of two intersecting tangent lines to a hyperbola are negative reciprocals of each other. This means that if one slope is m, the other slope will be -1/m.

## 5. How do you write the equation of the tangent line to a hyperbola in standard form?

The standard form of the equation of a tangent line to a hyperbola is y = mx + b, where m is the slope of the tangent line and b is the y-intercept. This form is useful for easily identifying the slope and y-intercept of the tangent line.

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