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

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

The curve $r=a / 1+\cos \theta$ ) intersects orthogonally with the curve $\begin{array}{llll}\text { LA) } r=b / 1-\cos \theta) & \text { (B) } r-b /(1+\sin \theta) & \text { (C) } r=b\left(1+\sin ^{2} \theta\right) & \left.\text { (D) } r=b / 1+\cos ^{2} \theta\right)\end{array}$ (V.T.U., 2010)

The curve $r=a / 1+\cos \theta$ ) intersects orthogonally with the curve
$\begin{array}{llll}\text { LA) } r=b / 1-\cos \theta) & \text { (B) } r-b /(1+\sin \theta) & \text { (C) } r=b\left(1+\sin ^{2} \theta\right) & \left.\text { (D) } r=b / 1+\cos ^{2} \theta\right)\end{array}$
(V.T.U., 2010)

Key Concepts

Intersection of Curves in Polar Coordinates

Finding the points where two polar curves intersect involves solving their equations simultaneously for the radius and angle. Once these intersection points are determined, the slopes of the curves at these points are computed to verify conditions such as orthogonality.

Differentiation in Polar Coordinates

When working with curves defined in polar form, obtaining the slope of a tangent line requires differentiating implicitly using the chain rule. The derivative dy/dx for a curve where r is given as a function of ? can be computed by differentiating the relations x = r cos(?) and y = r sin(?) and then applying the quotient rule.

Orthogonality of Curves

Two curves intersect orthogonally if their tangent lines at the point of intersection are perpendicular. In terms of slopes, this means that the product of the derivatives (slopes) of the two curves is -1. This condition is crucial in determining whether curves meet at right angles.

Polar Coordinates

Polar coordinates provide a way to represent points in a plane using a distance from a fixed origin and an angle from a fixed direction. This system is particularly useful for dealing with curves that exhibit radial symmetry or are easier to describe as a function of an angle rather than Cartesian x and y coordinates.

Conversion Between Polar and Cartesian Coordinates

Converting between polar and Cartesian coordinates allows one to apply techniques from Cartesian calculus to problems initially framed in polar form. The relationships x = r cos(?) and y = r sin(?) are key to transforming the representation of curves and are often used to compute slopes and tangents.

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