The curve r=a A 1+cosθ) intersects orthogonally with the curve (A) r=b / 1-cosθ) (B)

step 1 Here looking at the graph, we can see that this is a limouson with inner loop. So, this graph is

The curve r=a A 1+cosθ) intersects orthogonally with the...

The curve $r=a A 1+\cos \theta$ ) intersects orthogonally with the curve $\begin{array}{lll}\text { (A) } r=b / 1-\cos \theta) & \text { (B) } r=b /(1+\operatorname{lin} \theta) & \text { (C) } r=b\left(1+\sin ^{2} \theta\right)\end{array}$ $\begin{array}{ll} \left.\text { (D) } r=b / 11+\cos ^{2} \theta\right) & \left(\mathrm{V}, T, U_{4}, 2010\right)\end{array}$

The curve $r=a A 1+\cos \theta$ ) intersects orthogonally with the curve $\begin{array}{lll}\text { (A) } r=b / 1-\cos \theta) & \text { (B) } r=b /(1+\operatorname{lin} \theta) & \text { (C) } r=b\left(1+\sin ^{2} \theta\right)\end{array}$
$\begin{array}{ll} \left.\text { (D) } r=b / 11+\cos ^{2} \theta\right) & \left(\mathrm{V}, T, U_{4}, 2010\right)\end{array}$

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Key Concepts

Orthogonal Intersection of Curves

Two curves intersect orthogonally if their tangents at the point of intersection are perpendicular, meaning the product of their slopes is -1. This condition is used to determine relationships between families of curves, such as finding which curves form orthogonal trajectories with a given polar curve.

Differentiation of Polar Curves

When analyzing curves expressed in polar coordinates, computing the derivative dy/dx requires differentiating r with respect to ? and using the relationships x = r cos ? and y = r sin ?. This process allows one to determine the slope of the tangent at any point on the polar curve.

Polar Coordinate System

This concept involves representing points in the plane using a distance from the origin (r) and an angle (?) from a reference direction. It is essential for describing curves in polar form, like r = f(?), and understanding the geometry and behavior of such curves.

Slope Formula in Polar Coordinates

The slope of a polar curve can be found using the formula dy/dx = (dr/d? sin ? + r cos ?) / (dr/d? cos ? ? r sin ?). This formula is fundamental when studying the tangents and normals to curves given in polar form, which is crucial for understanding intersections between curves.

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