NCERT Class 11 Mathematics Solutions: Chapter 11 –Conic Sections Miscellaneous Exercise 11 Part 3
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Formula of Ellipse
1. An arch is in the form of a semi-ellipse. It is wide and high at the centre. Find the height of the arch at a point from one end.
Answer:
Since the height and width of the arc from the centre is and 8 m respectively, it is clear that the length of the major axis is , while the length of the semi-minor axis is .
The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is taken along the x-axis. Hence, the semi-ellipse can be diagrammatically represented as:
The Centre of the Ellipse
The equation of the semi-ellipse will be of the form where is the semi-major axis.
Accordingly,
So, the equation of the semi-ellipse is
Where,
Consider be a point on the major axis such that .
Draw .
The x-coordinate of point is
On substituting the value of with in equation (1)
(approx)
So, the height of the arch at a point 1 from one end is approximately
2. A rod of length moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point on the rod, which is from the end in contact with the axis.
Answer:
Consider be the rod making an angle with be the point on it such that .
Then,
From , draw .
AB be the Rod Making an Angle (Theta) with OX and P (X,Y)
In
In
Since,
So, the equation of the locus of point on the rod is