Find the equation of the tangent to the ellipse x^2+3 y^2=76 at each of the given points. Write

Find the equation of the tangent to the ellipse x^2+3 y^2=76...

Key Concepts

Point-Slope Form

The point-slope form is a way of writing the equation of a straight line when a point on the line and the slope of the line are known. It is generally given by the formula y - y? = m(x - x?), where (x?, y?) is the point on the line and m is the slope. This method is efficient for writing the equation of a tangent line once the slope at the point of tangency has been determined.

Implicit Differentiation

Implicit differentiation is a method used to differentiate equations where the variable y is defined implicitly as a function of x rather than explicitly. This technique involves differentiating both sides of the equation with respect to x and then solving for dy/dx, which represents the slope of the tangent line at any point on the curve. It is particularly useful for curves defined by equations like those of ellipses.

Tangent Line

A tangent line to a curve at a given point is a straight line that just touches the curve at that point and has the same direction as the curve's immediate path. In geometric and calculus contexts, the tangent line provides the best linear approximation of the curve near that point. Determining the tangent involves calculating the slope of the curve at the point of contact.

Ellipse

An ellipse is a type of conic section that results from slicing a cone with a plane at an angle. It can be defined algebraically by an equation of the form Ax² + By² = C, where A, B, and C are constants, which represents a closed, symmetric curve. Understanding its standard properties, such as axes, vertices, and symmetry, is essential for analyzing its geometry and for solving related problems like finding tangents.

Transcript

00:01 In problem 33, we need to find the equations of the tangent lines to the ellipse x squared plus 3y squared equal to 76 at these three different points.

00:11 And we're going to use the equation x1x over a squared plus y1y over b squared, and that's going to be equal to 1.

00:24 So let's go ahead and put our equation of our ellipse in standard form, which is going to be x squared over 76, and then plus 3 y squared over 76 equal to 1.

00:41 I normally wouldn't leave this 3 up in the numerator, but i don't like having fractions in the denominator, and for the specific application that we're doing in this problem, it doesn't really matter if i leave that 3 in the numerator.

00:54 So all we have to do now is just plug in all of these three different values into the equation that we're going to be using.

01:04 So let's begin in part 8, where x1 value is going to be 8, and it's x over 76, and it's going to be plus our y1 value, which is 2.

01:13 And remember, i left that 3 in the numerator, so that has to stay there times y over 76 equal to 1.

01:20 Now i'm going to go ahead and i'm going to multiply everything by 76.

01:26 So i get 8x plus 6y equal to 76, or 6y equal to negative 8x plus 76, and then i'm going to divide each side by 6, so i get y equal to negative 4 thirds x plus 38 thirds.

01:48 It does want us to put this in the y is equal to mx plus b format.

01:53 So that's why i did this simplification or this rearranging here.

01:57 I wouldn't know if i'd quite call it simplification.

02:01 That would be part a.

02:02 Now we can go ahead into part b, which is going to be the exact same process...

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