NCERT Class 11-Math's: Exemplar Chapter –11 Conic Sections Part 4

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Question 8:

An equilateral triangle is inscribed in the parabola whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

Answer:

As shown in the figure APQ denotes the equilateral triangle with its equal sides of length (say).

Equilateral Triangle

Equilateral Triangle

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Here AR

Also, .

Thus are the coordinates of the point P lying on the parabola .

Therefore, .

Thus, is the required length of the side of the equilateral triangle inscribed in the parabola .

Question 9:

Find the equation of the ellipse which passes through the point and has eccentricity , with x-axis as its major axis and centre at the origin.

Answer:

Let be the equation of the ellipse passing through the point .

Therefore, we have .

Or

Or

Or

Again

Hence, the required equation of the ellipse is

Or

Question 10:

Find the equation of the hyperbola whose vertices are and one of the directories is .

Answer:

As the vertices are on the and their middle point is the origin, the equation is of the type .

Here , vertices are and directrices are given by

Thus and so which gives

Consequently, the required equation of the hyperbola is

Objective Type Questions

Each of the examples from 11 to 16, has four possible options, out of which one is correct. Choose the correct answer from the given four options (M.C.Q.)

Question 11:

The equation of the circle in the first quadrant touching each coordinate axis at a distance of one unit from the origin is:

(A)

(B)

(C)

(C)

Answer:

The correct choice is (A), since the equation can be written as which represents a circle touching both the axes with its centre and radius one unit.

Question 12:

The equation of the circle having centre and passing through the point of intersection of the lines and is

(A)

(B)

(C)

(D)

Answer:

The correct option is (A).

The point of intersection of and are , i.e., the point

Therefore, the radius is and hence the equation of the circle is given by

or .