Find the equation of the ellipse whose focus is (1, - 2), directrix is 3x – 2y + 1 = 0 and
eccentricity is 1/√2.
Answers
EXPLANATION.
Equation of ellipse whose focus = (1,-2).
Directrix of an ellipse = 3x - 2y + 1 = 0.
Eccentricity of an ellipse = 1/√2.
As we know that,
If the general equation of an ellipse whose focus is (h, k) and the directrix is the line ax + by + c = 0. and the eccentricity will be e =
⇒ SP = e PM.
⇒ (SP)² = (e)²(PM)².
⇒ (x₁ - h)² + (y₁ - k)² = e²(ax₁ + by₁ + c)²/a² + b², e < 1.
Using this concept in equation, we get.
⇒ (x - 1)² + [y - (-2)]² = (1/√2)² (3x - 2y + 1)²/[(3)² + (2)²].
⇒ (x - 1)² + (y + 2)² = (1/2)(3x - 2y + 1)²/9 + 4.
⇒ (x - 1)² + (y + 2)² = 1/26 (3x - 2y + 1)².
As we know that,
Formula of :
⇒ (x - y)² = x² + y² - 2xy.
⇒ (x + y)² = x² + y² + 2xy.
⇒ (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx.
Using this formula in equation, we get.
⇒ x² + 1 - 2x + y² + 4 + 4y = 1/26 (3x - 2y + 1)².
⇒ 26[x² + y² - 2x + 4y + 5] = (9x² + 4y² + 1 - 12xy - 4y + 6x).
⇒ (26x² + 26y² - 52x + 104y + 130) = 9x² + 4y² - 12xy - 4y + 6x + 1).
⇒ 26x² + 26y² - 52x + 104y + 130 - 9x² - 4y² + 12xy + 4y - 6x - 1 = 0.
⇒ 17x² + 22y² - 58x + 108y + 129 = 0.
MORE INFORMATION.
Director Circle.
The equation of the director circle of the ellipse x²/a² + y²/b² = 1 is x² + y² = a² + b².
Diameter.
If y = mx + c represents a system of parallel chords of the ellipse x²/a² + y²/b² = 1 then the equation of the diameter is y = - b²x/a²m.
Conjugate diameter.
Two diameters are said to be conjugate when each bisects all chords parallel to the other.
If m₁ , m₂ be the slope of the conjugate diameters of an ellipse x²/a² + y²/b² = 1 then = m₁ m₂ = -b²/a².