Ellipse

Ellipse is expressed by equation 9x² + 25y² - 54x - 100y - 44 = 0. Find the length of primary and secondary axes, eccentricity, and coordinates of the center of the ellipse.

Result

o₁ =  10
o₂ =  6
x₀ =  3
y₀ =  2
e =  4

Solution:

$(\dfrac{ x-x_{ 0 } }{ a } )^2+(\dfrac{ y-y_{ 0 } }{ b } )^2 = 1 \ \\ (9x^2-54x)+(25y^2-100y) = 44 \ \\ 9 \cdot \ (x^2-6x)+25(y^2-4y) = 44 \ \\ \ \\ x^2-6x+9 = (x-3)^2 \ \\ y^2-4y+4 = (y-2)^2 \ \\ \ \\ 9 \cdot \ (x^2-6x+9)+25(y^2-4y+4) = 44+9 \cdot \ 9+4 \cdot \ 25 \ \\ \ \\ 9 \cdot \ (x-3)^2+25 \cdot \ (y-2)^2 = 225 \ \\ \dfrac{ 9 }{ 225 } \cdot \ (x-3)^2+\dfrac{ 25 }{ 225 } \cdot \ (y-2)^2 = 1 \ \\ \ \\ a = \sqrt{ \dfrac{ 225 }{ 9 } } = 5 \ \\ b = \sqrt{ \dfrac{ 225 }{ 25 } } = 3 \ \\ \ \\ o_{ 1 } = 2 \cdot \ a = 2 \cdot \ 5 = 10$

$o_{ 2 } = 2 \cdot \ b = 2 \cdot \ 3 = 6$

$x_{ 0 } = 3$

$y_{ 0 } = 2$

$e = \sqrt{ a^2-b^2 } = \sqrt{ 5^2-3^2 } = 4$

Try another example

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:

Be the first to comment!

Next similar math problems:

1. Circle - AG
   Find the coordinates of circle and its diameter if its equation is: ?
2. Equation of circle
   find an equation of the circle with indicated properties: a. center at (-3,5), diameter 20. b. center at origin and diameter 16.
3. Calculate
   Calculate the length of a side of the equilateral triangle with an area of 50cm².
4. Triangle KLM
   In the rectangular triangle KLM, where is hypotenuse m (sketch it!) find the length of the leg k and the height of triangle h if hypotenuse's segments are known mk = 5cm and ml = 15cm
5. Center of line segment
   Calculate the distance of the point X [1,3] from the center of the line segment x = 2-6t, y = 1-4t ; t is .
6. Median
   In triangle ABC is given side a=10 cm and median ta= 13 cm and angle gamma 90°. Calculate length of the median tb.
7. Vector 7
   Given vector OA(12,16) and vector OB(4,1). Find vector AB and vector |A|.
8. Circle annulus
   There are 2 concentric circles in the figure. Chord of larger circle 10 cm long is tangent to the smaller circle. What are does annulus have?
9. Triangle IRT
   In isosceles right triangle ABC with right angle at vertex C is coordinates: A (-1, 2); C (-5, -2) Calculate the length of segment AB.
10. Triangle ABC
    In a triangle ABC with the side BC of length 2 cm The middle point of AB. Points L and M split AC side into three equal lines. KLM is isosceles triangle with a right angle at the point K. Determine the lengths of the sides AB, AC triangle ABC.
11. Medians and sides
    Triangle ABC in the plane Oxy; are the coordinates of the points: A = 2.7 B = -4.3 C-6-1 Try calculate lengths of all medians and all sides.
12. Center
    Calculate the coordinates of the circle center: ?
13. Theorem prove
    We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?
14. Isosceles IV
    In an isosceles triangle ABC is |AC| = |BC| = 13 and |AB| = 10. Calculate the radius of the inscribed (r) and described (R) circle.
15. Euclid2
    In right triangle ABC with right angle at C is given side a=27 and height v=12. Calculate the perimeter of the triangle.
16. ABS CN
    Calculate the absolute value of complex number -15-29i.
17. Vertices of RT
    Show that the points P1 (5,0), P2 (2,1) & P3 (4,7) are the vertices of a right triangle.