# Math Assignment Help With Equations Of Tangent And Normal To An Ellipse

## 7.6.8 Equations of tangent and normal to an ellipse:

**Theorem:** The equation of tangent to the ellipse

*x*^{2} + y^{2} ‗ 1

a^{2} b^{2}

at a point (*x*1, y1) is

*xx*1 + yy1 ‗ 1

a^{2} b^{2}

and the equation of normal to the ellipse is *x*^{2} + y^{2} ‗ 1; at point (*x*1, y1) is

a^{2} b^{2}

(y – y1) ‗ (*x* – *x*1)

(y1/b^{2}) (*x*1/ a^{2})

**Proof:** Let P(*x*1, y1) is a given point on the ellipse *x<up>*^{2} + y^{2} ‗ 1 and Q(*x*2, y2) be a point on it.

a^{2} b^{2}

then the equation of the chord PQ is

y – y1 ‗ y2 – y1

*x* – *x*1 *x*2 – *x*1 …..(i)

as point p(*x*1, y1) and (*x*2, y2) lie on the given ellipse, therefore;

*x*1^{2} + y1^{2} ‗ 1 and *x*2^{2} + y2^{2}

a2 b2 a2 b2

On subtraction we get

*x*2^{2} – *x*1^{2} + y2^{2} – y1^{2} ‗ 1

a^{2} b^{2}

**10 x – 9y + 77 = 0**

The equation of the normal at (-5, 3)

(77/2) [ *x* – (-5)] /-5 = (77/3) (y – 3)/3

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### Following are some of the topics in Conic Sections-Parabola, Hyperbola And Ellipse in which we provide help:

- Introduction To Conic Sections
- Eccentricity
- Parameters Of Conic Section
- Parabola
- Forms Of Parabola
- General Equation Of Parabola
- Position Of A Point With Respect To Parabola
- Equation Of A Tangent And Normal At A Point
- Condition Of Tangency
- Point Of Contact Of A Line And A Parabola
- Ellipse
- Ellipse Properties

- Vertical Form Of Ellipse
- General Equation Of Ellipse
- Position Of A Point With Respect To An Ellipse
- Eccentric Angle
- Equations Of Tangent And Normal To An Ellipse
- Condition Of A Line To Touch An Ellipse
- Hyperbola
- Forms Of Hyperbola
- General Equation Of Hyperbola
- Tangent And Normal At A Point On Hyperbola
- Intersection Of A Line And Hyperbola
- Equation Of A Tangent From A Point Outside The Hyperbola

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