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Finding Equation of Ellipse from given 2 Foci and Major Axis Length

  1. Given a Real Ellipse having Coordinates of Center \((x_c,y_c)\), Coordinates of 2 Foci \((x_{f1},y_{f1})\) and \((x_{f2},y_{f2})\) and Length of Major Axis \(L=2a\), the Equation of the Real Ellipse can be found out as follows

    As per definition of Real Ellipse, the Sum of the Distance from 2 Foci to Any Point \((x,y)\) on the Ellipse is equal to the Length of it's Major Axis. Hence

    \(\sqrt{{(x-x_{f1})}^2 + {(y-y_{f1})}^2} + \sqrt{{(x-x_{f2})}^2 + {(y-y_{f2})}^2} =L=2a\)

    \(\Rightarrow \sqrt{{(x-x_{f1})}^2 + {(y-y_{f1})}^2} = 2a - \sqrt{{(x-x_{f2})}^2 + {(y-y_{f2})}^2}\)   ...(1)

    Squaring Both Sides of equation (1) above we get

    \({(x-x_{f1})}^2 + {(y-y_{f1})}^2 = 4a^2 + {(x-x_{f2})}^2 + {(y-y_{f2})}^2 - 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_{f2})}^2}\)

    \(\Rightarrow 4a^2 + {(x-x_{f2})}^2 + {(y-y_{f2})}^2 - {(x-x_{f1})}^2 - {(y-y_{f1})}^2 = 4a \sqrt{{(x-x_{f2})}^2 + {(y-y_{f2})}^2}\)   ...(2)

    Squaring Both Sides of equation (2) above we get

    \({(4a^2 + {(x-x_{f2})}^2 + {(y-y_{f2})}^2 - {(x-x_{f1})}^2 - {(y-y_{f1})}^2)}^2 = 16a^2 ({(x-x_{f2})}^2 + {(y-y_{f2})}^2)\)

    \(\Rightarrow {(4a^2 + {(x-x_{f2})}^2 + {(y-y_{f2})}^2 - {(x-x_{f1})}^2 - {(y-y_{f1})}^2)}^2 -16a^2 ({(x-x_{f2})}^2 + {(y-y_{f2})}^2)=0\)   ...(3)

    Expanding and Simplifying equation (3) above we get

    \((4{x_{f1}}^2+4{x_{f2}}^2-8{x_{f1}}{x_{f2}}-16a^2)x^2 + (4{y_{f1}}^2+4{y_{f2}}^2-8{y_{f1}}{y_{f2}}-16a^2) y^2 +(8{x_{f1}}{y_{f1}}-8{x_{f1}}{y_{f2}}+8{x_{f2}}{y_{f2}}-8{y_{f1}}{x_{f2}})xy\)
    \(+ (16a^2{x_{f1}}+16a^2{x_{f2}}-4{x_{f1}}^3-4{x_{f2}}^3+4{x_{f1}}{x_{f2}}^2+4{x_{f1}}{y_{f2}}^2-4{x_{f1}}{y_{f1}}^2+4{x_{f2}}{x_{f1}}^2+4{x_{f2}}{y_{f1}}^2-4{x_{f2}}{y_{f2}}^2)x\)
    \(+ (16a^2{y_{f1}}+16a^2{y_{f2}}-4{y_{f1}}^3-4{y_{f2}}^3+4{y_{f1}}{y_{f2}}^2+4{y_{f1}}{x_{f2}}^2-4{y_{f1}}{x_{f1}}^2+4{y_{f2}}{y_{f1}}^2+4{y_{f2}}{x_{f1}}^2-4{y_{f2}}{x_{f2}}^2)y\)
    \(+ (16a^4+{x_{f1}}^4+{y_{f1}}^4+{x_{f2}}^4+{y_{f2}}^4-8a^2{x_{f1}}^2-8a^2{y_{f1}}^2-8a^2{x_{f2}}^2-8a^2{y_{f2}}^2+2{x_{f1}}^2{y_{f1}}^2+2{x_{f2}}^2{y_{f2}}^2-2{x_{f1}}^2{x_{f2}}^2-2{y_{f1}}^2{x_{f2}}^2-2{x_{f1}}^2{y_{f2}}^2-2{y_{f1}}^2{y_{f2}}^2)=0\)   ...(4)

    The equation (4) above gives the Equation of Real Ellipse having Coordinates of 2 Foci at \((x_{f1},y_{f1})\) and \((x_{f2},y_{f2})\) and Length of Major Axis \(L=2a\) if \(L > F\), where \(F\) is Distance Between 2 Foci.

    Please note that the equation (4) above can also be used for Finding Equation of Hyperbola from given 2 Foci and Transverse Axis Length.
  2. The Ellipse represented by equation (4) gets converted to an \(X\)-Major Ellipse on setting \(y_{f1}=y_{f2}=y_f\) as given by the following calculations

    \((4{x_{f1}}^2+4{x_{f2}}^2-8{x_{f1}}{x_{f2}}-16a^2)x^2 + (4{y_f}^2+4{y_f}^2-8{y_f}{y_f}-16a^2) y^2 +(8{x_{f1}}{y_f}-8{x_{f1}}{y_f}+8{x_{f2}}{y_f}-8{y_f}{x_{f2}})xy\)
    \(+ (16a^2{x_{f1}}+16a^2{x_{f2}}-4{x_{f1}}^3-4{x_{f2}}^3+4{x_{f1}}{x_{f2}}^2+4{x_{f1}}{y_f}^2-4{x_{f1}}{y_f}^2+4{x_{f2}}{x_{f1}}^2+4{x_{f2}}{y_f}^2-4{x_{f2}}{y_f}^2)x\)
    \(+ (16a^2{y_f}+16a^2{y_f}-4{y_f}^3-4{y_f}^3+4{y_f}{y_f}^2+4{y_f}{x_{f2}}^2-4{y_f}{x_{f1}}^2+4{y_f}{y_f}^2+4{y_f}{x_{f1}}^2-4{y_f}{x_{f2}}^2)y\)
    \(+ (16a^4+{x_{f1}}^4+{y_f}^4+{x_{f2}}^4+{y_f}^4-8a^2{x_{f1}}^2-8a^2{y_f}^2-8a^2{x_{f2}}^2-8a^2{y_f}^2+2{x_{f1}}^2{y_f}^2+2{x_{f2}}^2{y_f}^2-2{x_{f1}}^2{x_{f2}}^2-2{y_f}^2{x_{f2}}^2-2{x_{f1}}^2{y_f}^2-2{y_f}^2{y_f}^2)=0\)

    \(\Rightarrow (4({x_{f1}}^2+{x_{f2}}^2-2{x_{f1}}{x_{f2}})-16a^2)x^2 + (8{y_f}^2-8{y_f}^2-16a^2) y^2 + (16a^2{x_{f1}}+16a^2{x_{f2}}-4{x_{f1}}^3-4{x_{f2}}^3+4{x_{f1}}{x_{f2}}^2+4{x_{f2}}{x_{f1}}^2)x\)
    \( + (32a^2{y_f}-4{y_f}^3-4{y_f}^3+4{y_f}^3+4{y_f}^3)y +(16a^4+{x_{f1}}^4+{x_{f2}}^4-2{x_{f1}}^2{x_{f2}}^2+2{y_f}^4-2{y_f}^4-16a^2{y_f}^2-8a^2({x_{f1}}^2+{x_{f2}}^2))=0\)

    \(\Rightarrow (4{(x_{f1}-x_{f2})}^2-16a^2)x^2 -16a^2y^2 + (16a^2(x_{f1}+x_{f2})-4x_{f1}({x_{f1}}^2-{x_{f2}}^2) +4x_{f2}({x_{f1}}^2-{x_{f2}}^2))x\)
    \( + 32a^2y_fy +(16a^4+{({x_{f1}}^2-{x_{f2}}^2)}^2-16a^2{y_f}^2-8a^2({x_{f1}}^2+{x_{f2}}^2))=0\)

    \(\Rightarrow (4{(x_{f1}-x_{f2})}^2-16a^2)x^2 -16a^2y^2 + (16a^2(x_{f1}+x_{f2})-4x_{f1}(x_{f1}-x_{f2})(x_{f1}+x_{f2}) +4x_{f2}(x_{f1}-x_{f2})(x_{f1}+x_{f2}))x\)
    \( + 32a^2y_fy +(16a^4+{((x_{f1}-x_{f2})(x_{f1}+x_{f2}))}^2-16a^2{y_f}^2-8a^2({x_{f1}}^2+{x_{f2}}^2))=0\)   ...(5)

    Now, we know that

    \(x_{f1}-x_{f2}=2c\)   ...(6)

    \(\Rightarrow {(x_{f1}-x_{f2})}^2=4c^2 \hspace{.5cm}\Rightarrow {x_{f1}}^2+{x_{f2}}^2-2{x_{f1}}{x_{f2}}=4c^2 \hspace{.5cm}\Rightarrow {x_{f1}}^2+{x_{f2}}^2=4c^2+2{x_{f1}}{x_{f2}}\)   ...(7)

    Also

    \(\frac{x_{f1}+x_{f2}}{2}=x_c\hspace{.5cm}\Rightarrow x_{f1}+x_{f2}=2x_c\)   (where \((x_c,y_c)\) are Coordinates of Center of Ellipse) ...(8)

    \(\Rightarrow {(x_{f1}+x_{f2})}^2=4{x_c}^2 \hspace{.5cm}\Rightarrow {x_{f1}}^2+{x_{f2}}^2+2{x_{f1}}{x_{f2}}=4{x_c}^2 \hspace{.5cm}\Rightarrow {x_{f1}}^2+{x_{f2}}^2=4{x_c}^2-2{x_{f1}}{x_{f2}}\)   ...(9)

    Adding equations (7) and (9) we get

    \( 2({x_{f1}}^2+{x_{f2}}^2)=4c^2+2{x_{f1}}{x_{f2}}+4{x_c}^2-2{x_{f1}}{x_{f2}}\hspace{.5cm}\Rightarrow 2({x_{f1}}^2+{x_{f2}}^2)=4(c^2+{x_c}^2)\hspace{.5cm}\Rightarrow ({x_{f1}}^2+{x_{f2}}^2)=2(c^2+{x_c}^2)\)   ...(10)

    Now, putting the values of \(x_{f1}-x_{f2}\), \(x_{f1}+x_{f2}\) and \({x_{f1}}^2+{x_{f2}}^2\) from equations (6), (8) and (10) in equation (5) we get

    \((4{(2c)}^2-16a^2)x^2 -16a^2y^2 + (16a^2(2x_c)-4x_{f1}(2c)(2x_c) +4x_{f2}(2c)(2x_c))x+ 32a^2y_fy +(16a^4+{((2c)(2x_c))}^2-16a^2{y_f}^2-8a^2(2(c^2+{x_c}^2)))=0\)

    \(\Rightarrow (16c^2-16a^2)x^2 -16a^2y^2 + (32a^2x_c -16x_{f1}cx_c +16x_{f2}cx_c)x + 32a^2y_fy +(16a^4+16c^2{x_c}^2-16a^2{y_f}^2-16a^2{x_c}^2-16a^2c^2)=0\)

    \(\Rightarrow -16(a^2-c^2)x^2 -16a^2y^2 + (32a^2x_c -16cx_c(x_{f1} -x_{f2}))x + 32a^2y_fy +(16a^4-16a^2c^2+16c^2{x_c}^2-16a^2{x_c}^2-16a^2{y_f}^2)=0\)

    \(\Rightarrow -16(a^2-c^2)x^2 -16a^2y^2 + (32a^2x_c -16cx_c(2c))x + 32a^2y_fy +(16a^2(a^2-c^2) - 16{x_c}^2(a^2-c^2) -16a^2{y_f}^2)=0\)

    \(\Rightarrow -16(a^2-c^2)x^2 -16a^2y^2 + (32a^2x_c -32c^2x_c)x + 32a^2y_fy +(16a^2(a^2-c^2) - 16{x_c}^2(a^2-c^2) -16a^2{y_f}^2)=0\)

    \(\Rightarrow -16(a^2-c^2)x^2 -16a^2y^2 + 32x_c (a^2 -c^2)x + 32a^2y_fy +(16a^2(a^2-c^2) - 16{x_c}^2(a^2-c^2) -16a^2{y_f}^2)=0\)   ...(11)

    Now, for any Ellipse \(a^2-c^2=b^2\). Substituting it in equation (11) above we get

    \(-16b^2x^2 -16a^2y^2 + 32b^2x_cx + 32a^2y_fy + 16a^2b^2 - 16b^2{x_c}^2 -16a^2{y_f}^2=0\)   ...(12)

    Dividing the equation (12) above by -16 on Both Sides we get

    \(b^2x^2 + a^2y^2 - 2b^2x_cx - 2a^2y_fy + b^2{x_c}^2 + a^2{y_f}^2 - a^2b^2=0\)   ...(13)

    Also for \(X\)-Major Ellipse \(y_f=y_c\). Hence,

    \(b^2x^2 + a^2y^2 - 2b^2x_cx - 2a^2y_cy + b^2{x_c}^2 + a^2{y_c}^2 - a^2b^2=0\)   ...(14)

    The equation (14) above gives the equation of \(X\)-Major Real Ellipse.
  3. The Ellipse represented by equation (4) gets converted to an \(Y\)-Major Ellipse on setting \(x_{f1}=x_{f2}=x_f\) as given by the following calculations

    \((4{x_f}^2+4{x_f}^2-8{x_f}{x_f}-16a^2)x^2 + (4{y_{f1}}^2+4{y_{f2}}^2-8{y_{f1}}{y_{f2}}-16a^2) y^2 +(8{x_f}{y_{f1}}-8{x_f}{y_{f2}}+8{x_f}{y_{f2}}-8{y_{f1}}{x_f})xy\)
    \(+ (16a^2{x_f}+16a^2{x_f}-4{x_f}^3-4{x_f}^3+4{x_f}{x_f}^2+4{x_f}{y_{f2}}^2-4{x_f}{y_{f1}}^2+4{x_f}{x_f}^2+4{x_f}{y_{f1}}^2-4{x_f}{y_{f2}}^2)x\)
    \(+ (16a^2{y_{f1}}+16a^2{y_{f2}}-4{y_{f1}}^3-4{y_{f2}}^3+4{y_{f1}}{y_{f2}}^2+4{y_{f1}}{x_f}^2-4{y_{f1}}{x_f}^2+4{y_{f2}}{y_{f1}}^2+4{y_{f2}}{x_f}^2-4{y_{f2}}{x_f}^2)y\)
    \(+ (16a^4+{x_f}^4+{y_{f1}}^4+{x_f}^4+{y_{f2}}^4-8a^2{x_f}^2-8a^2{y_{f1}}^2-8a^2{x_f}^2-8a^2{y_{f2}}^2+2{x_f}^2{y_{f1}}^2+2{x_f}^2{y_{f2}}^2-2{x_f}^2{x_f}^2-2{y_{f1}}^2{x_f}^2-2{x_f}^2{y_{f2}}^2-2{y_{f1}}^2{y_{f2}}^2)=0\)   ...(1)

    \(\Rightarrow (8{x_f}^2-8{x_f}^2-16a^2) x^2 + (4({y_{f1}}^2+{y_{f2}}^2-2{y_{f1}}{y_{f2}})-16a^2)y^2 + (32a^2{x_f}-4{x_f}^3-4{x_f}^3+4{x_f}^3+4{x_f}^3)x\)
    \( + (16a^2{y_{f1}}+16a^2{y_{f2}}-4{y_{f1}}^3-4{y_{f2}}^3+4{y_{f1}}{y_{f2}}^2+4{y_{f2}}{y_{f1}}^2)y +(16a^4+{y_{f1}}^4+{y_{f2}}^4-2{y_{f1}}^2{y_{f2}}^2+2{x_f}^4-2{x_f}^4-16a^2{x_f}^2-8a^2({y_{f1}}^2+{y_{f2}}^2))=0\)

    \(\Rightarrow -16a^2x^2 -(4{(y_{f1}-y_{f2})}^2-16a^2)y^2 + 32a^2x_fx + (16a^2(y_{f1}+y_{f2})-4y_{f1}({y_{f1}}^2-{y_{f2}}^2) +4y_{f2}({y_{f1}}^2-{y_{f2}}^2))y\)
    \(+(16a^4+{({y_{f1}}^2-{y_{f2}}^2)}^2-16a^2{x_f}^2-8a^2({y_{f1}}^2+{y_{f2}}^2))=0\)

    \(\Rightarrow -16a^2x^2 + (4{(y_{f1}-y_{f2})}^2-16a^2)y^2 + 32a^2x_fx + (16a^2(y_{f1}+y_{f2})-4y_{f1}(y_{f1}-y_{f2})(y_{f1}+y_{f2}) +4y_{f2}(y_{f1}-y_{f2})(y_{f1}+y_{f2}))y\)
    \( +(16a^4+{((y_{f1}-y_{f2})(y_{f1}+y_{f2}))}^2-16a^2{x_f}^2-8a^2({y_{f1}}^2+{y_{f2}}^2))=0\)   ...(15)

    Now, we know that

    \(y_{f1}-y_{f2}=2c\)   ...(16)

    \(\Rightarrow {(y_{f1}-y_{f2})}^2=4c^2 \hspace{.5cm}\Rightarrow {y_{f1}}^2+{y_{f2}}^2-2{y_{f1}}{y_{f2}}=4c^2 \hspace{.5cm}\Rightarrow {y_{f1}}^2+{y_{f2}}^2=4c^2+2{y_{f1}}{y_{f2}}\)   ...(17)

    Also

    \(\frac{y_{f1}+y_{f2}}{2}=y_c\hspace{.5cm}\Rightarrow y_{f1}+y_{f2}=2y_c\)   (where \((x_c,y_c)\) are Coordinates of Center of Ellipse) ...(18)

    \(\Rightarrow {(y_{f1}+y_{f2})}^2=4{y_c}^2 \hspace{.5cm}\Rightarrow {y_{f1}}^2+{y_{f2}}^2+2{y_{f1}}{y_{f2}}=4{y_c}^2 \hspace{.5cm}\Rightarrow {y_{f1}}^2+{y_{f2}}^2=4{y_c}^2-2{y_{f1}}{y_{f2}}\)   ...(19)

    Adding equations (17) and (19) we get

    \( 2({y_{f1}}^2+{y_{f2}}^2)=4c^2+2{y_{f1}}{y_{f2}}+4{y_c}^2-2{y_{f1}}{y_{f2}}\hspace{.5cm}\Rightarrow 2({y_{f1}}^2+{y_{f2}}^2)=4(c^2+{y_c}^2)\hspace{.5cm}\Rightarrow ({y_{f1}}^2+{y_{f2}}^2)=2(c^2+{y_c}^2)\)   ...(20)

    Now, putting the values of \(y_{f1}-y_{f2}\), \(y_{f1}+y_{f2}\) and \({y_{f1}}^2+{y_{f2}}^2\) from equations (16), (18) and (20) in equation (15) we get

    \(-16a^2x^2 + (4{(2c)}^2-16a^2)y^2 + 32a^2x_fx + (16a^2(2y_c)-4y_{f1}(2c)(2y_c) +4y_{f2}(2c)(2y_c))x +(16a^4+{((2c)(2y_c))}^2-16a^2{x_f}^2-8a^2(2(c^2+{y_c}^2)))=0\)

    \(\Rightarrow -16a^2x^2 +(16c^2-16a^2)y^2 + 32a^2x_fx + (32a^2y_c -16y_{f1}cy_c +16y_{f2}cy_c)y +(16a^4+16c^2{y_c}^2-16a^2{x_f}^2-16a^2{y_c}^2-16a^2c^2)=0\)

    \(\Rightarrow -16a^2x^2 -16(a^2-c^2)y^2 + 32a^2x_fx + (32a^2y_c -16cy_c(y_{f1} -y_{f2}))y +(16a^4-16a^2c^2+16c^2{y_c}^2-16a^2{y_c}^2-16a^2{x_f}^2)=0\)

    \(\Rightarrow -16a^2x^2 -16(a^2-c^2)y^2 + 32a^2x_fx + (32a^2y_c -16cy_c(2c))y +(16a^2(a^2-c^2) - 16{y_c}^2(a^2-c^2) -16a^2{x_f}^2)=0\)

    \(\Rightarrow -16a^2x^2 -16(a^2-c^2)y^2 + 32a^2x_fx + (32a^2y_c -32c^2y_c)y +(16a^2(a^2-c^2) - 16{y_c}^2(a^2-c^2) -16a^2{x_f}^2)=0\)

    \(\Rightarrow -16a^2x^2 -16(a^2-c^2)y^2 + 32a^2x_fx + 32y_c (a^2 -c^2)y + (16a^2(a^2-c^2) - 16{y_c}^2(a^2-c^2) -16a^2{x_f}^2)=0\)   ...(21)

    Now, for any Ellipse \(a^2-c^2=b^2\). Substituting it in equation (21) above we get

    \(-16a^2x^2 -16b^2y^2 + 32a^2x_fx + 32b^2y_cy + 16a^2b^2 - 16b^2{y_c}^2 -16a^2{x_f}^2=0\)   ...(22)

    Dividing the equation (22) above by -16 on Both Sides we get

    \(a^2x^2 + b^2y^2 - 2a^2x_fx - 2b^2y_cy + a^2{x_f}^2 + b^2{y_c}^2 - a^2b^2=0\)   ...(23)

    Also for \(Y\)-Major Ellipse \(x_f=x_c\). Hence,

    \(a^2x^2 + b^2y^2 - 2a^2x_cx - 2b^2y_cy + a^2{x_c}^2 + b^2{y_c}^2 - a^2b^2=0\)   ...(24)

    The equation (24) above gives the equation of \(Y\)-Major Real Ellipse.
Related Calculators
Conic from Foci and Major/Transverse Axis Length Calculator
Related Topics
Finding Equation of Ellipse from a given Focus, a Vertex and Eccentricity,    Finding Equation of Ellipse from given Adjacent Focus, Directrix and Eccentricity,    Finding Parametric Equations for Axis Aligned and Rotated Ellipse,    Introduction to Ellipse and Imaginary Ellipse,    General Quadratic Equations in 2 Variables and Conic Sections
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