Question

Assertion :The curve $y=k{x}^2$, (k is any arbitrary constant), intersect the curve $x^2+2y^2=2k$ at right angle. Reason: The above curves trace orthogonal trajectories.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

The equation of family of curves is given by
$y=k{x}^2$ (i)

$\therefore$ On differentiating equation (i) w.r to x, we get
$\frac{dy}{dx}=2kx$ (ii)

From (i) & (ii) eliminating k we have
$y=\left(\frac{1}{2x}\frac{dy}{dx}\right)x^2$
$\Rightarrow$ $2y=x.\frac{dy}{dx}$ (iii)

which is the differential equation of the family of curves given by equation (i)

The differential equation of the orthogonal trajectory in equation (i) is obtained by

replacing $\frac{dy}{dx}$ by $-\frac{dx}{dy}$ in the equation (iii)

so, we get $2y=-x\frac{dx}{dy}$
$\Rightarrow$ $2ydy=-xdx$ $\Rightarrow$ $y^2=\frac{-x^2}{2}+k$

$x^2+2y^2=2k$ (equation of family of orthogonal trajectories).