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ORTHOGONAL TRAJECTORY – MCQ- LEVEL-3

ORTHOGONAL TRAJECTORY – MCQ- LEVEL-3

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Question 1 of 10

1. Question
The orthogonal trajectory of family of curves ${\alpha ^{n - 1}}y = {x^n}$ are given by $( \alpha$ is arbitrary constant).
- $x^{n} + n^{2} y$ = constant
- $ny^{2} + x^{2}$ = constant
- $n^{2} x + y^{n}$ = constant
- $n^{2} x - y^{n}$ = constant
Correct

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Question 2 of 10

2. Question
If 2x(1 – y) = K and g(x, y) = L are orthogonal family of curves, where K and L are constant then g(x, y) is
- $x^{2} + 2y - y^{2}$
- $2y (1 - x)$
- $x^{2} + 2x - y^{2}$
- $x^{2} + 2y + y^{2}$
Correct

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Question 3 of 10

3. Question
If the differential equation of the family of curves $f(r,\theta ,c) = 0$ is $\phi \left( {r,\theta ,\frac{{dr}}{{d\theta }}} \right) = 0$ then the differential equation of orthogonal trajectory would be
- $\phi \left( {r,\theta ,{r^2}\dfrac{{d\theta }}{{dr}}} \right) = 0$
- $\phi \left( {r,\theta, - {r^2}\dfrac{{d\theta }}{{dr}}} \right) = 0$
- $\phi \left( {r,\theta ,{r^2}\dfrac{{dr}}{{d\theta }}} \right) = 0$
- $\phi \left( {r,\theta ,{\theta ^2}\dfrac{{dr}}{{d\theta }}} \right) = 0$
Correct

Incorrect
Question 4 of 10

4. Question
The orthogonal trajectory for system of curve $\left( {\dfrac{{dy}}{{dx}}} \right)^2 = \dfrac{a}{x}$ is
- $\left( {{x^{3/2}} + {c^{3/2}}} \right) = \dfrac{9}{4}ay$
- ${x^{3/2}} - {c^{3/2}} = \dfrac{3}{2}\sqrt a y$
- ${({x^{3/2}} + {c^{3/2}})^2} = \dfrac{8}{7}{a^2}{y^2}$
- ${({x^{3/2}} + {c^{3/2}})^2} = \frac{9}{4}a{y^2}$
Correct

Incorrect
Question 5 of 10

5. Question
Orthogonal trajectory of $r\theta = a$ is
- $r = c{e^{\dfrac{{ - {\theta ^2}}}{2}}}$
- $r = c{e^{\dfrac{{{\theta ^2}}}{2}}}$
- ${r^2} = {c^2}{e^{{\theta ^2}}}$
- ${r^2} = {c^2}{e^{{\theta ^2}/2}}$
Correct

Incorrect
Question 6 of 10

6. Question
Find the equation of system of orthogonal trajectories of the parabolas $r = \dfrac{{2a}}{{1 + \cos \theta }},$ where a is the parameter.
- $r = \dfrac{{2b}}{{1 + \cos 2\theta }}$
- $r = \dfrac{{2b}}{{1 - \cos 2\theta }}$
- $r = \dfrac{{2b}}{{1 - \cos \theta }}$
- $r = \dfrac{{2b}}{{1 - \sin \theta }}$
Correct

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Question 7 of 10

7. Question
The orthogonal trajectories of the family of curves y = 3x³ + x + c are
- $y = 3tan^{-1} 3x + c_{1}$
- $y = 3x^{3} + x + c_{1}$
- $y = tan^{-1} x + c_{1}$
- $9y + 3tan^{-1} 3x = c_{1}$
Correct

Incorrect
Question 8 of 10

8. Question
The condition for which ax² + by² = 1 and a₁x² + b₁y² = 1, shall cut orthogonally is
- $aa_{1} (b_{1} - b) + bb_{1} (a - a_{1}) = 0$
- $aa_{1} (b_{1} - b) - bb_{1} (a - a_{1}) = 0$
- $aa_{1} (a - a_{1} ) + bb_{1} (b_{1} - b) = 0$
- $aa_{1} (a - a_{1} ) - bb_{1} (b_{1} - b) = 0$
Correct

Incorrect
Question 9 of 10

9. Question
The orthogonal trajectories for family of parabolas is the
- family parabolas
- family of ellipse
- family circle
- family of hyperbola
Correct

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Question 10 of 10

10. Question
The orthogonal trajectories for the family of circle with centre at x-axis is
- family of circle with centre on x-axis
- family of ellipses
- family circles with centre on y axis
- Any type of family of circles
Correct

Incorrect

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