Tangent of a circle is a line which touches the circle at only one point and normal is a line perpendicular to the tangent and passing through the point of contact.

**Equations of tangent and normal at a point ***P ***on a given circle**

Tangent of a circle is a line which touches the circle at only
one point and normal is a line perpendicular to the tangent and passing through
the point of contact.

Let *P*(*x*_{1} , *y*_{1} ) and *Q*(*x*_{2}
, *y*_{2} ) be two points on the circle *x*^{2} + *y*^{2}
+ 2*gx *+ 2 *fy *+ *c *= 0 .

Therefore,

*x _{1}*

and *x*_{2}^{2}
+ *y*_{2}^{2} + 2*gx*_{2} + 2*fy*_{2}+ c = 0 ……..(2)

(2) - (1) gives

Therefore, slope of PQ = - (x_{1} + x_{2} + 2*g*) / ( y_{1} + y_{2}
+ 2*f* )

When Q → P, the chord PQ becomes tangent at P

Slope of tangent is

Hence, the equation of tangent is y – y_{1} = Simplifying

* yy _{1} +
fy – y_{1}^{2} - fy_{1} + xx_{1} - x_{1}^{2}
+ gx - gx*

* xx _{1}
+ yy_{1} + gx + fy - *(

Since (*x*_{1} , *y*_{1} ) is a point
on the circle, we have *x*_{1}^{2} + *y*_{1}^{2} + 2*gx*_{1} + 2 *fy*_{1}
+ *c *= 0

Therefore, -(*x*_{1}^{2} + *y*_{1}^{2} + *gx *+ *fy *) = *gx*_{1}* *+ *fy*_{1}* *+ *c *(2)

Hence, substituting (2) in (1), we get the equation of tangent
at (*x*_{1} , *y*_{1} ) as

*xx*_{1} + *yy*_{1} + *g*(*x *+ *x*_{1}
) + *f *( *y *+ *y*_{1} ) + *c *= 0 .

Hence, the equation of normal is

(* y – y*_{1} ) =
[ ( *y*_{1} + *f *) / ( *x*_{1} + *g* )
] ( *x* – *x*_{1} )

Þ ( *y *- *y*_{1}
)( *x*_{1} + *g *) = ( *y*_{1} + *f *)(
*x *- *x*_{1} )

Þ *x*_{1}
( *y *- *y*_{1} ) + *g *( *y *- *y*_{1}
) = *y*_{1} ( *x *- *x*_{1} ) + *f *( *x *-
*x*_{1} )

Þ *yx*_{1}
- *xy*_{1} + *g *( *y *- *y*_{1} ) - *f *(
*x *- *x*_{1} ) = 0 .

(1) The equation of tangent at ( *x*_{1} , *y*_{1}
) to the circle *x*^{2}+*y*^{2} = a^{2} is *xx*_{1}
+ *yy*_{1} = *a*^{2}.

(2) The equation of normal at ( *x*_{1} , *y*_{1}
) to the circle *x*^{2}+*y*^{2} = a^{2} is *xy*_{1}
- *yx*_{1} = 0 .

(3) The normal passes through the centre of the circle.

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12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II : Equations of tangent and normal at a point P on a given circle |

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