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Home » Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
Application of Derivatives | Class 12 | Exercise 1 | Maths | NCERT Exemplar
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.

It’s seen that the given bends are condition of two circles.

    \[2x\text{ }=\text{ }y2\text{ }\ldots \text{ }..\text{ }\left( 1 \right)\]

and

    \[2xy\text{ }=\text{ }k\text{ }\ldots \text{ }..\text{ }\left( 2 \right)\]

We realize that, two circles converge symmetrically if the point between the digressions attracted to the two circles at the place of their convergence is 90o.

Presently, separating conditions (1) and (2) w.r.t. t, we get

NCERT Exemplar Solutions Class 12 Mathematics Chapter 6 - 19

NCERT Exemplar Solutions Class 12 Mathematics Chapter 6 - 20

Presently, settling conditions (1) and (2), we have

    \[y\text{ }=\text{ }k/2x\text{ }\left[ From\text{ }\left( 2 \right) \right]\]

Placing the worth of y in condition (1),

    \[2x\text{ }=\text{ }\left( k/2x \right)2\Rightarrow 2x\text{ }=\text{ }k2/4x2\]

    \[8x3\text{ }=\text{ }k2\]

    \[8\left( 1 \right)3\text{ }=\text{ }k2\]

    \[k2\text{ }=\text{ }8\]

Subsequently, the necessary condition is

    \[k2\text{ }=\text{ }8.\]

i

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