Summary

Videos

References

An ellipse is the set of all the points in a plane the sum of whose distances from two fixed points is constant.

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2}

Two foci, two vertices, centre, major axis and minor axis of an ellipse are as shown in the figure.

In an ellipse,

*c*^{2} = *a*^{2} - *b*^{2},^{ }

Where,

*c* = Distance of a focus from the centre

*a* = Length of semi-major axis

*b* = Length of semi-minor axis

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2} = k

Let K be the constant equal to the sum of the distances of any point on the ellipse from its foci.

Vertex A lies on the ellipse,

⇒ AF_{1} + AF_{2} = k ……(1)

AF_{1} = AO - F_{1}O

⇒ AF_{1} = *a* - *c*

AF_{2} = AO + OF_{2}

⇒ AF_{2} = *a* + *c*

⇒ AF_{1} + AF_{2} = *a* - *c* + *a* + *c* = k

⇒ k = 2*a*

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2} = 2*a*

Thus, the sum of the distances of any point on an ellipse from its foci is equal to the length of the major axis, or two times the semi-major axis of the ellipse.

Eccentricity is a measure of the deviation of a conic section from being a circle. The eccentricity of an ellipse is denoted by 'e' and is equal to the ratio of the distance of a focus from the centre to the length of the semi-major axis of the ellipse. Eccentricity of an ellipse (*e*) = *c*/*a*

Squaring both sides, we get

*e*^{2} = *c*^{2}/*a*^{2}

⇒ *e*^{2} = (*a*^{2} - *b*^{2})/*a*^{2} (Since *c*^{2} = *a*^{2} - *b*^{2})

⇒ *e*^{2} = 1- *b*^{2}/*a*^{2}

⇒ *a*^{2}*e*^{2} = *a*^{2}- *b*^{2}

⇒ *b*^{2}* = a*^{2 }*- a*^{2}*e*^{2}

⇒ *b*^{2} = *a*^{2} (1- *e*^{2})

Some standard equations that are satisfied by all the points lying on an ellipse

If we consider an ellipse with its vertex at the origin and its major axis along the X- or the Y-axis, then there are two distinct possibilities.

Case I: Centre (0, 0), major axis along the *X*-axis.

Case II: Centre (0, 0), major axis along the *Y*-axis.

Consider a point P on the ellipse with the coordinates X, Y.

PF_{1} + PF_{2} = 2*a* ……(1)

Distance between the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) = √(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2} ^{ }

PF_{1} = √[(*x* - (-*c*)]^{2} + (*y* - 0)^{2}

⇒ PF_{1} = √(*x* + *c*)^{2} + *y*^{2} ……(2)

PF_{2} = √(*x* - *c*)^{2} + (*y* - 0)^{2}

⇒ PF_{2} = √(*x* - *c*)^{2} + *y*^{2} ……(3)

From equations (1), (2) and (3),

√(*x* + *c*)^{2} + *y*^{2} + √(*x* - *c*)^{2} + *y*^{2} = 2*a*

⇒ √(*x* + *c*)^{2} + *y*^{2} = 2*a* - √(*x* - *c*)^{2} + *y*^{2}

Squaring both sides, we get

(*x* + *c*)^{2} + *y*^{2} = [2*a* - √(*x* - *c*)^{2} + *y*^{2}]^{2}

(*x* + *c*)^{2} + *y*^{2} = (2*a*)^{2} - 2(2a)(√(*x* - *c*)^{2} + *y*^{2}) + (√(*x* - *c*)^{2} + *y*^{2})^{2}

(*x* + *c*)^{2} + *y*^{2} = 4*a*^{2} - 4*a*√(*x* - *c*)^{2} + *y*^{2} + (*x* - *c*)^{2} + *y*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = (*x* - *c*)^{2} - (*x* + *c*)^{2} + 4*a*^{2}

= *x*^{2} + *c*^{2} - 2*xc* -*x*^{2} - *c*^{2} - 2*xc* + 4*a*^{2}

= 4*a*^{2} - 4*xc*

⇒ *a*√(*x* - *c*)^{2} + *y*^{2} = *a*^{2} - *xc*

⇒ *a*√(*x* - *c*)^{2} + *y*^{2} = *a*^{2} - *xc*

⇒ √(*x* - *c*)^{2} + *y*^{2} = *a* - (c/a) *x*

Squaring both sides, we get

(*x* - *c*)^{2} + *y*^{2} = *[a* - (c/a) *x] ^{2}*

(*x* - *c*)^{2} + *y*^{2} = *(a) ^{2}* - 2.a.(c/a)

(*x* - *c*)^{2} + *y*^{2} = *a*^{2} + (c^{2}/a^{2}*) x* - 2*cx*

⇒*x*^{2} + *c*^{2} - 2*cx* + *y*^{2} = *a*^{2} + (c^{2}/a^{2}*)**x*^{2} - 2*cx*

⇒*x*^{2} + *c*^{2} + *y*^{2} = *a*^{2} + (c^{2}/a^{2}*)**x*^{2}

⇒*x*^{2} - (c^{2}/a^{2}*)**x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒(1 - (c^{2}/a^{2}))*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒( (a^{2}-c^{2})/a^{2})*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2} ……(4)

⇒( b^{2}/a^{2})*x*^{2} + *y*^{2} = *b*^{2 } (Since *a*^{2} - *c*^{2} = *b*^{2})

Multiplying both sides by 1/*b*^{2},

x^{2}/a^{2} + y^{2}/b^{2} = 1 ……(5)

⇒ PF_{1} = √(*x* + *c*)^{2} + *b*^{2} (1 - x^{2}/a^{2} From equations ...(2) and (5))

= √ *x*^{2} + *c*^{2} + 2*cx* + *b*^{2} ^{ }- b^{2}x^{2}/a^{2}

= √ *x*^{2}(1- b^{2}/a^{2} + 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √ *x*^{2}((a^{2}-b^{2})/a^{2}) + 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √(c^{2}/a^{2})x^{2}+ 2*cx* + *a*^{2} (Since *c*^{2} = *a*^{2} - *b*^{2})

= √(a + c/a)x^{2}

⇒ PF_{1} = a + (c/a)x

⇒ PF_{2} = √(*x* - *c*)^{2} + *b*^{2} (1 - x^{2}/a^{2}) (From equation (3),

= √ *x*^{2} + *c*^{2} - 2*cx* + *b*^{2} ^{ }- b^{2}x^{2}/a^{2}

= √ *x*^{2}(1- b^{2}/a^{2} - 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √ *x*^{2}(a^{2}-b^{2}/a^{2})- 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √(c^{2}/a^{2})x^{2}- 2*cx* + *a*^{2}

= √(a - *cx/a)*^{2}

⇒ PF_{2} = a - *cx/a*

⇒ PF_{1} + PF_{2} = a + *cx/a* + a - *cx/a* = 2*a*

The equation x^{2}/a^{2} + y^{2}/b^{2} = 1 is the standard equation for an ellipse with its centre lying at the origin and the major axis lying along the X-axis.

Similarly, the standard equation for an ellipse with its centre at the origin and the major axis along the Y-axis can be derived as x^{2}/b^{2} + y^{2}/a^{2} = 1.

These two equations are called the standard equations of an ellipse having its centre at the origin and the major axis along the X- or the Y-axis.

In an ellipse, the length of the major axis is always greater than the length of the minor axis.

If the denominator of the *x*^{2} term is greater than the denominator of the *y*^{2} term, then the major axis lies along the *X*-axis.

If the denominator of the *y*^{2} term is greater than the denominator of the *x*^{2} term, then the major axis lies along the *Y*-axis.

Both the standard equations of an ellipse contain even powers of X and Y.

An ellipse with its centre lying at the origin and the major axis along either the *X-* or the Y-axis is symmetrical about both the coordinate axes.

⇒ If (*x*, *y*) lies on the ellipse, then (-*x*, *y*), (*x*, -*y*) and (-*x*, -*y*) also lie on the ellipse.

**Length of Latus Rectum**

A line segment passing through the focus and perpendicular to the major axis with its end points lying on the curve of an ellipse is called the latus rectum of the ellipse.

Since an ellipse has two foci, it has two latus recta. Latus recta is the plural for latus rectum.

Consider an ellipse with its centre at the origin and the major axis along the X-axis and **Latus Recta** AB and CD.

Let AF_{2} = l

⇒ Coordinates of A are (*c*, l), where

*c* = Distance of focus F_{2} from centre O

Equation of the given ellipse is x^{2}/a^{2} + y^{2}/b^{2} = 1.

Sine A (*c*, l) lies on x^{2}/a^{2} + y^{2}/b^{2} = 1.

⇒ c^{2}/a^{2} + l^{2}/b^{2} = 1

⇒ l^{2} = b^{2}( 1 - c^{2}/a^{2} )

⇒ l^{2} = b^{2}( (a^{2} - c^{2})/a^{2} )

⇒ l^{2} = b^{2}( b^{2}/a^{2}) (Since *b*^{2} = *a*^{2} - *c*^{2})

⇒ l^{2} = b^{4}/a^{2}

⇒ l = b^{2}/a

AF_{2} = F_{2}B = b^{2}/a

⇒ AB = AF_{2} + F_{2}B = 2b^{2}/a

Length of latus rectum of an ellipse = 2b^{2}/a, where

*a* = Length of semi-major axis

*b* = Length of semi-minor axis.

An ellipse is the set of all the points in a plane the sum of whose distances from two fixed points is constant.

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2}

Two foci, two vertices, centre, major axis and minor axis of an ellipse are as shown in the figure.

In an ellipse,

*c*^{2} = *a*^{2} - *b*^{2},^{ }

Where,

*c* = Distance of a focus from the centre

*a* = Length of semi-major axis

*b* = Length of semi-minor axis

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2} = k

Let K be the constant equal to the sum of the distances of any point on the ellipse from its foci.

Vertex A lies on the ellipse,

⇒ AF_{1} + AF_{2} = k ……(1)

AF_{1} = AO - F_{1}O

⇒ AF_{1} = *a* - *c*

AF_{2} = AO + OF_{2}

⇒ AF_{2} = *a* + *c*

⇒ AF_{1} + AF_{2} = *a* - *c* + *a* + *c* = k

⇒ k = 2*a*

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2} = 2*a*

Thus, the sum of the distances of any point on an ellipse from its foci is equal to the length of the major axis, or two times the semi-major axis of the ellipse.

Eccentricity is a measure of the deviation of a conic section from being a circle. The eccentricity of an ellipse is denoted by 'e' and is equal to the ratio of the distance of a focus from the centre to the length of the semi-major axis of the ellipse. Eccentricity of an ellipse (*e*) = *c*/*a*

Squaring both sides, we get

*e*^{2} = *c*^{2}/*a*^{2}

⇒ *e*^{2} = (*a*^{2} - *b*^{2})/*a*^{2} (Since *c*^{2} = *a*^{2} - *b*^{2})

⇒ *e*^{2} = 1- *b*^{2}/*a*^{2}

⇒ *a*^{2}*e*^{2} = *a*^{2}- *b*^{2}

⇒ *b*^{2}* = a*^{2 }*- a*^{2}*e*^{2}

⇒ *b*^{2} = *a*^{2} (1- *e*^{2})

Some standard equations that are satisfied by all the points lying on an ellipse

If we consider an ellipse with its vertex at the origin and its major axis along the X- or the Y-axis, then there are two distinct possibilities.

Case I: Centre (0, 0), major axis along the *X*-axis.

Case II: Centre (0, 0), major axis along the *Y*-axis.

Consider a point P on the ellipse with the coordinates X, Y.

PF_{1} + PF_{2} = 2*a* ……(1)

Distance between the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) = √(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2} ^{ }

PF_{1} = √[(*x* - (-*c*)]^{2} + (*y* - 0)^{2}

⇒ PF_{1} = √(*x* + *c*)^{2} + *y*^{2} ……(2)

PF_{2} = √(*x* - *c*)^{2} + (*y* - 0)^{2}

⇒ PF_{2} = √(*x* - *c*)^{2} + *y*^{2} ……(3)

From equations (1), (2) and (3),

√(*x* + *c*)^{2} + *y*^{2} + √(*x* - *c*)^{2} + *y*^{2} = 2*a*

⇒ √(*x* + *c*)^{2} + *y*^{2} = 2*a* - √(*x* - *c*)^{2} + *y*^{2}

Squaring both sides, we get

(*x* + *c*)^{2} + *y*^{2} = [2*a* - √(*x* - *c*)^{2} + *y*^{2}]^{2}

(*x* + *c*)^{2} + *y*^{2} = (2*a*)^{2} - 2(2a)(√(*x* - *c*)^{2} + *y*^{2}) + (√(*x* - *c*)^{2} + *y*^{2})^{2}

(*x* + *c*)^{2} + *y*^{2} = 4*a*^{2} - 4*a*√(*x* - *c*)^{2} + *y*^{2} + (*x* - *c*)^{2} + *y*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = (*x* - *c*)^{2} - (*x* + *c*)^{2} + 4*a*^{2}

= *x*^{2} + *c*^{2} - 2*xc* -*x*^{2} - *c*^{2} - 2*xc* + 4*a*^{2}

= 4*a*^{2} - 4*xc*

⇒ *a*√(*x* - *c*)^{2} + *y*^{2} = *a*^{2} - *xc*

⇒ *a*√(*x* - *c*)^{2} + *y*^{2} = *a*^{2} - *xc*

⇒ √(*x* - *c*)^{2} + *y*^{2} = *a* - (c/a) *x*

Squaring both sides, we get

(*x* - *c*)^{2} + *y*^{2} = *[a* - (c/a) *x] ^{2}*

(*x* - *c*)^{2} + *y*^{2} = *(a) ^{2}* - 2.a.(c/a)

(*x* - *c*)^{2} + *y*^{2} = *a*^{2} + (c^{2}/a^{2}*) x* - 2*cx*

⇒*x*^{2} + *c*^{2} - 2*cx* + *y*^{2} = *a*^{2} + (c^{2}/a^{2}*)**x*^{2} - 2*cx*

⇒*x*^{2} + *c*^{2} + *y*^{2} = *a*^{2} + (c^{2}/a^{2}*)**x*^{2}

⇒*x*^{2} - (c^{2}/a^{2}*)**x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒(1 - (c^{2}/a^{2}))*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒( (a^{2}-c^{2})/a^{2})*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2} ……(4)

⇒( b^{2}/a^{2})*x*^{2} + *y*^{2} = *b*^{2 } (Since *a*^{2} - *c*^{2} = *b*^{2})

Multiplying both sides by 1/*b*^{2},

x^{2}/a^{2} + y^{2}/b^{2} = 1 ……(5)

⇒ PF_{1} = √(*x* + *c*)^{2} + *b*^{2} (1 - x^{2}/a^{2} From equations ...(2) and (5))

= √ *x*^{2} + *c*^{2} + 2*cx* + *b*^{2} ^{ }- b^{2}x^{2}/a^{2}

= √ *x*^{2}(1- b^{2}/a^{2} + 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √ *x*^{2}((a^{2}-b^{2})/a^{2}) + 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √(c^{2}/a^{2})x^{2}+ 2*cx* + *a*^{2} (Since *c*^{2} = *a*^{2} - *b*^{2})

= √(a + c/a)x^{2}

⇒ PF_{1} = a + (c/a)x

⇒ PF_{2} = √(*x* - *c*)^{2} + *b*^{2} (1 - x^{2}/a^{2}) (From equation (3),

= √ *x*^{2} + *c*^{2} - 2*cx* + *b*^{2} ^{ }- b^{2}x^{2}/a^{2}

= √ *x*^{2}(1- b^{2}/a^{2} - 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √ *x*^{2}(a^{2}-b^{2}/a^{2})- 2*cx* + *c*^{2} + *b*^{2} ^{ }

= √(c^{2}/a^{2})x^{2}- 2*cx* + *a*^{2}

= √(a - *cx/a)*^{2}

⇒ PF_{2} = a - *cx/a*

⇒ PF_{1} + PF_{2} = a + *cx/a* + a - *cx/a* = 2*a*

The equation x^{2}/a^{2} + y^{2}/b^{2} = 1 is the standard equation for an ellipse with its centre lying at the origin and the major axis lying along the X-axis.

Similarly, the standard equation for an ellipse with its centre at the origin and the major axis along the Y-axis can be derived as x^{2}/b^{2} + y^{2}/a^{2} = 1.

These two equations are called the standard equations of an ellipse having its centre at the origin and the major axis along the X- or the Y-axis.

In an ellipse, the length of the major axis is always greater than the length of the minor axis.

If the denominator of the *x*^{2} term is greater than the denominator of the *y*^{2} term, then the major axis lies along the *X*-axis.

If the denominator of the *y*^{2} term is greater than the denominator of the *x*^{2} term, then the major axis lies along the *Y*-axis.

Both the standard equations of an ellipse contain even powers of X and Y.

An ellipse with its centre lying at the origin and the major axis along either the *X-* or the Y-axis is symmetrical about both the coordinate axes.

⇒ If (*x*, *y*) lies on the ellipse, then (-*x*, *y*), (*x*, -*y*) and (-*x*, -*y*) also lie on the ellipse.

**Length of Latus Rectum**

A line segment passing through the focus and perpendicular to the major axis with its end points lying on the curve of an ellipse is called the latus rectum of the ellipse.

Since an ellipse has two foci, it has two latus recta. Latus recta is the plural for latus rectum.

Consider an ellipse with its centre at the origin and the major axis along the X-axis and **Latus Recta** AB and CD.

Let AF_{2} = l

⇒ Coordinates of A are (*c*, l), where

*c* = Distance of focus F_{2} from centre O

Equation of the given ellipse is x^{2}/a^{2} + y^{2}/b^{2} = 1.

Sine A (*c*, l) lies on x^{2}/a^{2} + y^{2}/b^{2} = 1.

⇒ c^{2}/a^{2} + l^{2}/b^{2} = 1

⇒ l^{2} = b^{2}( 1 - c^{2}/a^{2} )

⇒ l^{2} = b^{2}( (a^{2} - c^{2})/a^{2} )

⇒ l^{2} = b^{2}( b^{2}/a^{2}) (Since *b*^{2} = *a*^{2} - *c*^{2})

⇒ l^{2} = b^{4}/a^{2}

⇒ l = b^{2}/a

AF_{2} = F_{2}B = b^{2}/a

⇒ AB = AF_{2} + F_{2}B = 2b^{2}/a

Length of latus rectum of an ellipse = 2b^{2}/a, where

*a* = Length of semi-major axis

*b* = Length of semi-minor axis.