Find the equation of the tangent line to 9 x^2+16 y^2=52 through (2,-1) (A) -9 x+8 y-26=0 (B) 9

step 1 The first thing we can do is find D -Y over DX. We might be doing implicit differentiation here.

Find the equation of the tangent line to 9 x^2+16 y^2=52...

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Key Concepts

Ellipse

An ellipse is a conic section defined by a quadratic equation. Its geometric properties, such as axes lengths and symmetry, are crucial when analyzing how lines interact with it, especially when determining tangent lines.

Tangent Line

A tangent line to a curve is one that touches the curve at exactly one point and has the same slope as the curve at that point. It represents the instantaneous direction of the curve at the point of contact.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated. This is particularly useful for finding the slope of tangent lines for curves defined by implicit relationships.

Point-Slope Form

The point-slope form of a linear equation is used to express the equation of a line when a point on the line and its slope are known. It is a key tool for determining the explicit equation of any tangent line.

Tangency Condition

The tangency condition in the context of conic sections ensures that a given line touches the curve at exactly one point. This often involves substituting the line’s equation into the curve’s equation and setting the discriminant of the resulting quadratic equation to zero.

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