How do we find equation of tangent and normal to a circle at a given point?

Answer 1

Equation of tangent at #(x_1,y_1)# at circle #x^2+y^2+2gx+2fy+c=0# is #x x_1+yy_1+g(x+x_1)+f(y+y_1)+c=0# and equation of normal is #(y_1+f)x-(x_1+g)y-fx_1+gy_1=0#

Let the circle be #x^2+y^2+2gx+2fy+c=0#,

let us seek normal and tangent at #(x_1,y_1)#

(note that #x_1^2+y_1^2+2gx_1+2fy_1+c=0#) ...(A)

as the center of the circle is #(-g,-f)#,

as normal joins #(-g,-f)# and #(x_1,y_1)# its slope is #(y_1+f)/(x_1+g)#

and equation of normal is #y-y_1=(y_1+f)/(x_1+g)(x-x_1)#

i.e. #(x_1+g)y-x_1y_1-gy_1=(y_1+f)x-x_1y_1-fx_1#

or #(y_1+f)x-(x_1+g)y-fx_1+gy_1=0#

and as product of slopes of normal and tangent is #-1#,

slope of tangent is #-(x_1+g)/(y_1+f)# and its equation will be

#y-y_1=-(x_1+g)/(y_1+f)(x-x_1)#

or #(y-y_1)(y_1+f)+(x-x_1)(x_1+g)=0#

or #yy_1-y_1^2+fy-fy_1+x x_1-x_1^2+gx-gx_1=0#

or #x x_1+yy_1+gx+fy-(x_1^2+y_1^2+fy_1+gx_1)=0#

from (A) #x_1^2+y_1^2+fy_1+gx_1=-fy_1-gx_1-c#

Hence equation of tangent is #x x_1+yy_1+gx+fy+fy_1+gx_1+c=0#

or #x x_1+yy_1+g(x+x_1)+f(y+y_1)+c=0#

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email

Answer 2

Evelyn Arboleda

To find the equation of the tangent to a circle at a given point, we first need to know the center of the circle and the radius. Let the center of the circle be $(h, k)$ and the radius be $r$ .

Tangent: If the point of tangency is $(x_1, y_1)$ , then the equation of the tangent is $(y - y_1) = m(x - x_1)$ , where $m$ is the slope of the tangent and is given by $-\frac{1}{\text{slope of the radius}}$ .
Normal: The normal to the circle at a given point is perpendicular to the tangent at that point. So, the slope of the normal is the negative reciprocal of the slope of the tangent. The equation of the normal passing through $(x_1, y_1)$ is given by $(y - y_1) = (-\frac{1}{m})(x - x_1)$ , where $m$ is the slope of the tangent.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.