Chapter 11: Conic Sections Miscellaneous Exercise

We know that the origin of the coordinate plane is taken at the vertex of the parabolic reflector, where the axis of the reflector is along the positive x – axis.

Diagrammatic representation is as follows:

We know that the equation of the parabola is of the form y² = 4ax (as it is opening to the right)

Since, the parabola passes through point A(10, 5),

y² = 4ax

10² = 4a(5)

100 = 20a

a = 100/20

= 5

The focus of the parabola is (a, 0) = (5, 0), which is the mid – point of the diameter.

Hence, the focus of the reflector is at the mid-point of the diameter.

We know that the origin of the coordinate plane is taken at the vertex of the arch, where its vertical axis is along the positive y –axis.

Diagrammatic representation is as follows:

The equation of the parabola is of the form x² = 4ay (as it is opening upwards).

It is given that at base arch is 10 m high and 5 m wide.

So, y = 10 and x = 5/2 from the above figure.

It is clear that the parabola passes through point (5/2, 10)

So, x² = 4ay

(5/2)² = 4a(10)

4a = 25/(4×10)

a= 5/32

we know the arch is in the form of a parabola whose equation is x² = 5/8y

We need to find width, when height = 2 m.

To find x, when y = 2.

When, y = 2,

x² = 5/8 (2)

= 5/4

x = √(5/4)

= √5/2

AB = 2 × √5/2 m

= √5m

= 2.23 m (approx.)

Hence, when the arch is 2 m from the vertex of the parabola, its width is approximately 2.23 m.

We know that the vertex is at the lowest point of the cable. The origin of the coordinate plane is taken as the vertex of the parabola, while its vertical axis is taken along the positive y –axis.

Diagrammatic representation is as follows:

Here, AB and OC are the longest and the shortest wires, respectively, attached to the cable.

DF is the supporting wire attached to the roadways, 18 m from the middle.

So, AB = 30 m, OC = 6m, and BC = 50 m.

The equation of the parabola is x² = 4ay (as it is opening upwards).

The coordinates of point A are (50, 30 -6) = (50, 24)

Since, A(50, 24) is a point on the parabola.

y² = 4ax

(50)² = 4a(24)

a = (50×50)/(4×24)

= 625/24

Equation of the parabola, x² = 4ay = 4×(625/24)y or 6x² = 625y

The x – coordinate of point D is 18.

Hence, at x = 18,

6(18)² = 625y

y = (6×18×18)/625

= 3.11 (approx.)

Thus, DE = 3.11 m

DF = DE +EF = 3.11 m +6 m = 9.11 m

Hence, the length of the supporting wire attached to the roadway 18 m from the middle is approximately 9.11 m.

Since, the height and width of the arc from the centre is 2 m and 8 m, respectively, it is clear that the length of the major axis is 8 m, while the length of the semi- minor axis is 2 m.

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, Diagrammatic representation of semi- ellipse is as follows:

The equation of the semi – ellipse will be of the from x²/16 + y²/4 = 1, y ≥ 0 … (1

Let A be a point on the major axis such that AB = 1.5m.

Now draw AC ⊥ OB.

OA = (4 – 1.5)m = 2.5m

The x – coordinate of point C is 2.5

On substituting the value of x with 2.5 in equation (1), we get,

(2.5)²/16 + y²/4 = 1

6.25/16 + y²/4 = 1

y² = 4 (1 – 6.25/16)

= 4 (9.75/16)

= 2.4375

y = 1.56 (approx.)

So, AC = 1.56m

Hence, the height of the arch at a point 1.5m from one end is approximately 1.56m.

Let AB be the rod making an angle Ɵ with OX and P(x,y) be the point on it such that

AP = 3cm.

Diagrammatic representation is as follows:

Then, PB = AB – AP = (12 – 3) cm = 9cm [AB = 12cm]

From P, draw PQ ⊥ OY and PR ⊥ OX.

In ΔPBQ, cos θ = PQ/PB = x/9

Sin θ = PR/PA = y/3

we know that, sin² θ +cos² θ = 1,

So,

(y/3)² + (x/9)² = 1 or

x²/81 + y²/9 = 1

Hence, the equation of the locus of point P on the rod is x²/81 + y²/9 = 1

The given parabola is x² = 12y.

On comparing this equation with x² = 4ay, we get,

4a = 12

a = 12/4

= 3

The coordinates of foci are S(0,a) = S(0,3).

Now let AB be the latus rectum of the given parabola.

The given parabola can be roughly drawn as

At y = 3, x² = 12(3)

x² = 36

x = ±6

So, the coordinates of A are (-6, 3), while the coordinates of B are (6, 3)

Then, the vertices of ΔOAB are O(0,0), A (-6,3) and B(6,3).

By using the formula,

Area of ΔOAB = ½ [0(3-3) + (-6)(3-0) + 6(0-3)] unit²

= ½ [(-6) (3) + 6 (-3)] unit²

= ½ [-18-18] unit²

= ½ [-36] unit²

= 18 unit²

∴ Area of ΔOAB is 18 unit²

Let A and B be the positions of the two flag posts and P(x, y) be the position of the man.

So, PA + PB = 10.

We know that if a point moves in plane in such a way that the sum of its distance from two fixed point is constant, then the path is an ellipse and this constant value is equal to the length of the major axis of the ellipse.

Then, the path described by the man is an ellipse where the length of the major axis is 10m, while points A and B are the foci.

Now let us take the origin of the coordinate plane as the centre of the ellipse, and taking the major axis along the x- axis,

The diagrammatic representation of the ellipse is as follows:

The equation of the ellipse is in the form of x²/a² + y²/b² = 1, where ‘a’ is the semi-major axis.

So, 2a = 10

a = 10/2

= 5

Distance between the foci, 2c = 8

c = 8/2

= 4

By using the relation, c = √(a² – b²), we get,

4 = √(25 – b²)

16 = 25 – b²

b² = 25 -1

= 9

b = 3

Hence, equation of the path traced by the man is x²/25 + y²/9 = 1

Let us consider OAB be the equilateral triangle inscribed in parabola y² = 4ax.

Let AB intersect the x – axis at point C.

Diagrammatic representation of the ellipse is as follows:

Now let OC = k

From the equation of the given parabola, we have,

So, y² = 4ak

y = ±2√ak

The coordinates of points A and B are (k, 2√ak), and (k, -2√ak)

AB = CA + CB

= 2√ak + 2√ak

= 4√ak

Since, OAB is an equilateral triangle, OA² = AB².

Then,

k² + (2√ak)² = (4√ak)²

k² + 4ak = 16ak

k² = 12ak

k = 12a

Thus, AB = 4√ak = 4√(a×12a)

= 4√12a²

= 4√(4a×3a)

= 4(2)√3a

= 8√3a

Hence, the side of the equilateral triangle inscribed in parabola y² = 4ax is 8√3a.

Exercise 11.1 Solutions 15 Questions

Exercise 11.2 Solutions 12 Questions

Exercise 11.3 Solutions 20 Questions

Exercise 11.4 Solutions 15 Questions

Also explore – NCERT Class 11 Solutions