NCERT Class 11 Mathematics Solutions: Chapter 11 –Conic Sections Miscellaneous Exercise 11 Part 3

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In figure formula of ellipse is given.

Formula of Ellipse

In figure formula of ellipse is given.

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 origin of the coordinate plane is taken 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).

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