# Equation Of A Tangent And Normal At A Point

## 7.5.6 Equation of a tangent and normal at a point

**Theorem1 **the equation of tangent to the parabola y^{2} = 4a*x* at the point (*x1*, y1) is yy1 = 2a(*x + x1*) and the equation of normal at this point is **2a(y – y1) + y( x – x1) = 0**

**Proof: **let P(x1, y1) and Q(x2, y2) lie on the parabola y^{2} = 4ax, also we have y1^{2} = 4ax1, y2^{2} = 4ax2

The equation of chord

y – y1 ‗ y2 – y1

*x* – *x*1 *x*2 – *x*1 …..(i)

## Equation Of A Tangent And Normal At A Point Assignment Help By Online Tutoring and Guided Sessions at AssignmentHelp.Net

The point P and Q lie on the parabola y2 = 4a*x* we have y1^{2} = 4ax1 and y2^{2} = 4ax2

y2 – y1 4a

*x*2 – *x*1 y1 + y2 ….(ii)

using (i) and (ii) we get the equation of chord PQ as

y – y1 4a

*x* – *x*1 y1 + y2

When Q→ P , then *x*2 → *x*1 ,y2 → y1 and chord PQ tends to tangent P. hence the equation of tangent at P is given by

y – y1 2a

*x* – *x*1 y1

yy1 = 2a(*x* – *x*1) + y1^{2} = 2a(*x – x*1) + 4a*x*1

**yy1 = 2a( x + x1)**

this is the required equation of the tangent to the parabola.

The slope of the tangent is given by; **2a**

**y**1

Slope of the normal is given by; **-y1**

** 2a**

Hence the equation of the normal

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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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