Family of Curves

Q. If the solution curve of the differential equation $(2x-10y^3)dy+ydx=0,$ passes through the points $(0,1)$ and $(2,\beta )$, then $\beta$ is a root of the equation

1. $2y^5-y^2-2=0$
2. $y^5-y^2-1=0$
3. $y^5-2y-2=0$
4. $2y^5-2y-1=0$

Q. Equation of the parabola whose directrix is y = 2x – 9 and focus (-8, -2) is

1. 
   
2. 
   
3. 
4. None of these

Q.

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Q. Let $C_1$ be the curve obtained by solution of differential equation $2xy\frac{dy}{dx}=y^2-x^2,x>0$ and curve $C_2$ be the solution of the differential equation $\frac{2xy}{x^2-y^2}=\frac{dy}{dx}.$ If both curve passes through $(1,1),$ then the area enclosed by the curves $C_1$ and $C_2$ is equal to :

1. $(\frac{\pi }{2}-1)$ sq. units
2. $(\frac{\pi }{4}-1)$ sq. units
3. $(\pi -1)$ sq. units
4. $(\pi +1)$ sq. units

Q.

Find the area of the region bounded by the curves $y=x^2+2,y=x,x=0$ and x = 3.

Q. Differential equation of the family of parabolas whose vertex lie on the $x-$ axis and focus as origin is

1. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}+y=0$
2. $y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y=0$
3. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$
4. $y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}+y=0$

Q. Let $f\left(x\right)$ be a real valued continuous function satisfying
$f^3\left(x\right)-5f^2\left(x\right)+10f\left(x\right)-12\le 0$
and $f^2\left(x\right)-7f\left(x\right)+12\le 0.$
A ray of light coming along the curve $y=f\left(x\right)$ from positive direction of $x-$axis and strikes the inner surface of mirror $y=2\sqrt{3x}$. If $l$ is the length of the reflected ray contained within the parabola $y^2=12x,$ then the value of $12l$ is

Q. The differential equation representing the family of circles with their centres on $x-$axis and whose radius is equal to the distance from from $(-1,2)$ to the line $3x+4y-15=0,$ is given by $y^2[{\left(\frac{dy}{dx}\right)}^2+k]=4,$ then $k^2+5$ is equal to

Q. Equation of curve which passes through point $(1,1)$ and satisfies the differential equation $3x{y}^{2}dy=(x^2+2y^3)dx$ is

1. $y^3=x\ln \left(e{x}^2\right)$
2. $y=\ln \left(xe\right)$
3. $y^6=x^4\ln \left(x^{2}e\right)$
4. $y^3=x^2\ln \left(ex\right)$

Q. The absolute value of the constant term in the solution of the differential equation $y(2x^4+y)\frac{dy}{dx}=(1-4x{y}^2)x^2$,
if the curve passes through the center of the circle $x^2+y^2-6y=0$, is

Q. The equation of the directrix of the parabola with vertex at the origin and having the axis along $x$ − axis and a common tangent of slope $2$ with the circle $x^2+y^2=5$ is⁄are

1. $x=10$
2. $x=20$
3. $x=-10$
4. $x=-20$

Q. If a curve $y=f\left(x\right)$, passing through the point $(1,2)$ is the solution of the differential equation, $2x^{2}dy=(2xy+y^2)dx$, then $f\left(\frac{1}{2}\right)$ is equal to:

1. $\frac{-1}{1+{\log }_{e}2}$
2. $1+{\log }_{e}2$
3. $\frac{1}{1+{\log }_{e}2}$
4. $\frac{1}{1-{\log }_{e}2}$

Q. Equation of the parabola whose axis is horizontal and passing through the points $\left(-2,1\right),\left(1,2\right),\left(-1,3\right)$ is

Q. Equation of curve which passes through point $(1,1)$ and satisfies the differential equation $3x{y}^{2}dy=(x^2+2y^3)dx$ is

1. $y^3=x\ln \left(e{x}^2\right)$
2. $y^6=x^4\ln \left(x^{2}e\right)$
3. $y^3=x^2\ln \left(ex\right)$
4. $y=\ln \left(xe\right)$

Q.

There is a line and a parabola $y^2=4x$ on the xy plane. The line intersects the parabola at only 1 point. If one of the point lying on this line is (105, 1) , then which among the following points would also lie on the line.

1. (1, 1)

2. (105, 160)

3. (81, 0)

4. (-1, -25)

Q. The area bounded by the tangent and normal to the curve $y(6-x)=x^2$ at $(3,3)$ and the x-axis is

1. $5$
2. $6$
3. $15$
4. $3$

Q. The point (s) on the curve $y^3+3x^2=12y,$ where the tangent is vertical (i.e., parallel to the y-axis), is / true

1. $(\pm \frac{4}{\sqrt{3}},-2)$
2. $(0,0)$
3. $(\pm \frac{\sqrt{11}}{3},1)$
4. $(\pm \frac{4}{\sqrt{3}},2)$

Q.

The order of differential equation of family of curves given by $y=a_1(a_2+a_3)\cdot \cos (x+a_4)-a_5{e}^{x+a_6},$ is

1. $6$
2. $5$
3. $4$
4. $3$

Q. The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

1. $(x^2-y^2)\frac{dy}{dx}-2xy=0$
2. $(x^2-y^2)\frac{dy}{dx}+2xy=0$
3. $(x^2-y^2)\frac{dy}{dx}-xy=0$
4. $(x^2-y^2)\frac{dy}{dx}+xy=0$

Q. The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

1. $(x^2-y^2)\frac{dy}{dx}-2xy=0$
2. $(x^2-y^2)\frac{dy}{dx}+2xy=0$
3. $(x^2-y^2)\frac{dy}{dx}-xy=0$
4. $(x^2-y^2)\frac{dy}{dx}+xy=0$

Q. The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is

1. $(x^2-y^2)\frac{dy}{dx}-2xy=0$
2. $(x^2-y^2)\frac{dy}{dx}+2xy=0$
3. $(x^2-y^2)\frac{dy}{dx}-xy=0$
4. $(x^2-y^2)\frac{dy}{dx}+xy=0$

Q. Differential equation of the family of parabolas whose vertex lie on the $x-$ axis and focus as origin is

1. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}-y=0$
2. $y{\left(\frac{dy}{dx}\right)}^2+2x\frac{dy}{dx}+y=0$
3. $y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}-y=0$
4. $y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}+y=0$

Q. The first order differential equation of the family of circles with fixed radius $r$ and with centre on $x$ -axis is:

1. $x^2(\frac{dy}{dx}{)}^2+y^2=r^2$
2. $y^2(\frac{dy}{dx}{)}^2+y^2=r^2$
3. $(\frac{dy}{dx}{)}^2+y^2=r^2$
4. $y^2-(\frac{dy}{dx}{)}^2=r^2$

Q. Form the differential equation of the family of parabolas having vertex at origin and axis along positive $y$-axis.

Q. The differential equation of the family of circles passing through the fixed points $(a,0)$ and $(-a,0)$ is:

1. $y_1(y^2-x^2)+2xy+a^2=0$
2. none of these
3. $y_1{y}^2+xy+a^2{x}^2=0$
4. $y_1(y^2-x^2+a^2)+2xy=0$

Q. $y=x^2-6x+8$
The equation above represents a parabola in the $xy$-plane. Which of the following equivalent forms of the equation displays the $x$-intercepts of the parabola as constants or coefficients?

1. $y-8=x^2-6x$
2. $y+1=(x-3{)}^2$
3. $y=x(x-6)+8$
4. $y=(x-2)(x-4)$

Q. Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus recta being $4a$ and $4b$. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a

1. straight line if $a=b$
2. none of these
3. parabola if $a\ne b$
4. parabola for $a,b$

Q.