# Family of Curves

## Trending Questions

**Q.**If the solution curve of the differential equation (2x−10y3)dy+ydx=0, passes through the points (0, 1) and (2, β), then β is a root of the equation

- 2y5−y2−2=0
- y5−y2−1=0
- y5−2y−2=0
- 2y5−2y−1=0

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

- None of these

**Q.**

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

**Q.**Let C1 be the curve obtained by solution of differential equation 2xydydx=y2−x2, x>0 and curve C2 be the solution of the differential equation 2xyx2−y2=dydx. If both curve passes through (1, 1), then the area enclosed by the curves C1 and C2 is equal to :

- (π2−1) sq. units
- (π4−1) sq. units
- (π−1) sq. units
- (π+1) sq. units

**Q.**

Find the area of the region bounded by the curves y=x2+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

- y(dydx)2+2xdydx+y=0
- y(dydx)2+xdydx−y=0
- y(dydx)2+2xdydx−y=0
- y(dydx)2+xdydx+y=0

**Q.**Let f(x) be a real valued continuous function satisfying

f3(x)−5f2(x)+10f(x)−12≤0

and f2(x)−7f(x)+12≤0.

A ray of light coming along the curve y=f(x) from positive direction of x−axis and strikes the inner surface of mirror y=2√3x. If l is the length of the reflected ray contained within the parabola y2=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 y2[(dydx)2+k]=4, then k2+5 is equal to

**Q.**Equation of curve which passes through point (1, 1) and satisfies the differential equation 3xy2dy=(x2+2y3)dx is

- y3=xln(ex2)
- y=ln(xe)
- y6=x4ln(x2e)
- y3=x2ln(ex)

**Q.**The absolute value of the constant term in the solution of the differential equation y(2x4+y)dydx=(1−4xy2)x2,

if the curve passes through the center of the circle x2+y2−6y=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 x2+y2=5 is⁄are

- x=10
- x=20
- x=−10
- x=−20

**Q.**If a curve y=f(x), passing through the point (1, 2) is the solution of the differential equation, 2x2dy=(2xy+y2)dx, then f(12) is equal to:

- −11+loge 2
- 1+loge2
- 11+loge2
- 11−loge2

**Q.**Equation of the parabola whose axis is horizontal and passing through the points (−2, 1), (1, 2), (−1, 3) is

**Q.**Equation of curve which passes through point (1, 1) and satisfies the differential equation 3xy2dy=(x2+2y3)dx is

- y3=xln(ex2)
- y6=x4ln(x2e)
- y3=x2ln(ex)
- y=ln(xe)

**Q.**

There is a line and a parabola y2=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)

(105, 160)

(81, 0)

(-1, -25)

**Q.**The area bounded by the tangent and normal to the curve y(6−x)=x2 at (3, 3) and the x-axis is

- 5
- 6
- 15
- 3

**Q.**The point (s) on the curve y3+3x2=12y, where the tangent is vertical (i.e., parallel to the y-axis), is / true

- (±4√3, −2)
- (0, 0)
- (±√113, 1)
- (±4√3, 2)

**Q.**

The order of differential equation of family of curves given by y=a1(a2+a3)⋅cos(x+a4)−a5ex+a6, is

- 6
- 5
- 4
- 3

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

- (x2−y2)dydx−2xy=0
- (x2−y2)dydx+2xy=0
- (x2−y2)dydx−xy=0
- (x2−y2)dydx+xy=0

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

- (x2−y2)dydx−2xy=0
- (x2−y2)dydx+2xy=0
- (x2−y2)dydx−xy=0
- (x2−y2)dydx+xy=0

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

- (x2−y2)dydx−2xy=0
- (x2−y2)dydx+2xy=0
- (x2−y2)dydx−xy=0
- (x2−y2)dydx+xy=0

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

- y(dydx)2+2xdydx−y=0
- y(dydx)2+2xdydx+y=0
- y(dydx)2+xdydx−y=0
- y(dydx)2+xdydx+y=0

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

- x2(dydx)2+y2=r2
- y2(dydx)2+y2=r2
- (dydx)2+y2=r2
- y2−(dydx)2=r2

**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:

- y1(y2−x2)+2xy+a2=0
- none of these
- y1y2+xy+a2x2=0
- y1(y2−x2+a2)+2xy=0

**Q.**y=x2−6x+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?

- y−8=x2−6x
- y+1=(x−3)2
- y=x(x−6)+8
- 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

- straight line if a=b
- none of these
- parabola if a≠b
- parabola for a, b

**Q.**

Find the differential equation of all the circles which pass through the origin and whose centres lie on x-axis.