Check the orthogonality of the curves \(y^2\) = x and \(x^2\) = y.

Solution :

Solving the curves simultaneously we get points of intersection as (1, 1) and (0, 0).

At (1, 1) for first curve \(2y({dy\over dx})_1\) = 1  \(\implies\)  \(m_1\) = \(1\over 2\)

& for second curve 2x = \(({dy\over dx})_2\) \(\implies\)  \(m_2\) = 2

\(m_1m_2\) = -1 at (1, 1).

But at (0, 0) clearly x-axis & y-axis are their respective tangents hence they are orthogonal at (0, 0) but not at (1, 1). Hence these curves are not said to be orthogonal.

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