A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man

 ncert cbse 11th mathematics chapter 11 conic section

 

7. A man running a racecourse notes that the sum of the distances from the two flagposts from him is always 10 m and the distance between the flag posts is 8 m.
Find the equation of the posts traced by the man

let P be a point on the locus

If we take the flagposts as F and F'

we are given that PF + PF' =10m ( a constant )

FF' = 8m

Using properties of an ellipse , the locus is an ellipse with

foci at F and F'

and PF + PF' =2a , and FF' = 2c

so that 2a = 10  and 2c = 8

or a = 10/2 = 5 , so  a^2 = 25

c = 8/2 = 4

use (c^2) = (a^2) - (b^2)

(b^2) = (a^2) - (c^2)

(b^2) = (5^2) - (4^2) =25 -16 = 9

 (b^2) =9 note that this is (b^2) not just b

equation of ellipse is 

[(x^2) / (a^2)] + [(y^2) / (b^2)] = 1

 [(x^2) / (25)] + [(y^2) / (9)] = 1

 

8. An equilateral triangle is inscribed in the parabola (y^2) = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

 Let u be the side of the equilateral triangle

Given the vertex of the parabola O(0,0)  is also a vertex of the equilateral triangle

 If A and B are the other two vertices,  the axis of the parabola, x axis is a perpendicular bisector of AB.

let M be the point of intersection of AB and the x axis.

clearly OA = AB =OB = u

and BM =MA =(u/2) 

By symmetry of the parabola about the x axis, we can choose

the vertices A and  B as A( x,(u/2) ) and B( x,(-u/2) )

clearly MO = x

MA = (u/2) 

using pythagoras theorem in right angled triangle OAM

(x^2) = (u^2) - [(u/2)^2]

(x^2) = (u^2) - [(u^2)/ 4 ]

 (x^2) =[3(u^2)/ 4 ]

x = sqrt(3) * u /2

so that A( x,(u/2) ) changes to A( {sqrt(3) * u /2} , (u/2)  )

A lies on  (y^2) = 4 ax,

so  (u/2)^2 = 4a {sqrt(3) * u /2}

  (u^2) / 4 = 2a *sqrt(3)*u

since u cannot be zero

u = 8 * sqrt(3) *a

 

 

 

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ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

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2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

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3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

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4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

solution  

 5. A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

solution

Find the area of the triangle formed by the lines joining the vertex of the parabola
x^2 = 12y to the ends of its latus rectum.  

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7. A man running a racecourse notes that the sum of the distances from the two flagposts from him is always 10 m and the distance between the flag posts is 8 m.
Find the equation of the posts traced by the man

solution 

8. An equilateral triangle is inscribed in the parabola (y^2) = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

 solution 

 

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