Question
) Find the condition for the curves, and
intersecting orthogonally.
Answer :
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For two curves to intersect orthogonally, the product of their slopes at the point of intersection must be (-1). The first curve is (ax^2 + by^2 = 1). Differentiating implicitly with respect to (x): [ \frac{d}{dx}(ax^2 + by^2) = 0 \implies 2ax + 2by \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{ax}{by}. ] The second curve is (a'x^2 + b'y^2 = 1). Differentiating implicitly: [ 2a'x + 2b'y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{a'x}{b'y}. ] For orthogonal intersection, the product of slopes must satisfy: [ \left(-\frac{ax}{by}\right)\left(-\frac{a'x}{b'y}\right)
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