As we know that if a point moves in a plane in such a way that its distance from two-point remain constant then the path is an ellipse.
Now, According to the question,
the distance between the point from where the sum of the distance from a point is constant = 10
Now, the distance between the foci=8
Now, As we know the relation,
Hence the equation of the ellipse is,
Hence the path of...
Given, an equilateral triangle inscribed in parabola with the equation.
The one coordinate of the triangle is A(0,0).
Now, let the other two coordinates of the triangle are
and
Now, Since the triangle is equilateral,
The coordinates of the points of the equilateral triangle are,
So, the side of the triangle is
Given the parabola,
Comparing this equation with , we get
Now, As we know the coordinates of ends of latus rectum are:
So, the coordinates of latus rectum are,
Now the area of the triangle with coordinates (0,0),(6,3) and (-6,3)
Widht of the triangle = 2*6=12
Height of the triangle = 3
So The area =
Hence the required area is 18 unit square.
Let be the angle that rod makes with the ground,
Now, at a point 3 cm from the end,
At the point touching the ground
Now, As we know the trigonometric identity,
Hence the equation is,
The equation of the semi-ellipse will be of the form
Now, According to the question,
the length of major axis = 2a = 8
The length of the semimajor axis =2
Hence the equation will be,
Now, at point 1.5 cm from the end, the x coordinate is 4-1.5 = 2.5
So, the height at this point is
Hence the height of the required point is 1.56 m.
Given,
The width of the parabolic cable = 100m
The length of the shorter supportive wire attached = 6m
The length of the longer supportive wire attached = 30m
Since the rope opens towards upwards, the equation will be of the form
Now if we consider origin at the centre of the rope, the equation of the curve will pass through points, (50,30-6)=(50,24)
Hence the equation of the parabola...
Le the parabolic reflector opens towards the right.
So the equation of parabolic reflector will be,
Now, Since this curve will pass through the point (5,10) if we assume origin at the optical centre,
So
Hence, The focus of the parabola is,
.
Alternative Method,
As we know on any concave curve
Hence, Focus
.
Hence the focus is 5 cm right to the optical centre.