# A)find the equations of the tangent lines to the c...

a)find the equations of the tangent lines to the circlex^2+y^2=25 at the points where x = 4.
b) find the equations of the normal lines to this circle atthe same points . ( the normal line is perpendicular to the tangentline at that point)
c) at what points do the two normal lines intersect?
can i please get a detailed worked out solution, so i get theconcept and know how to do other problems on my own?i'd be verygrateful and give you an A.thanks!

## Answers ( 1 )

1. a)find the equations of the tangent lines to the circlex^2+y^2=25 at the points where x = 4.
x^2+y^2=25..........................1
Put x=4 in the above equation
u get y^2 = 25-16 = 9
This gives y = +3or - 3
There fore there will be two tangents and normals, one atx1,y1 = 4,3 andother at x2,y2=4,-3
Differentiate eqn 1, you get
2x+2y y' = 0 ========> y ' = -2x/2y =-x/y.......................2
Put x,y = 4,3 in eqn 2, uget m1 = slope of first tangent = -4/3
Put x,y = 4, -3 in eqn 2 toget m2= slope of second tangent = 4/3
eqn of tangent is given by
y-Y = m(x-X) where X,Y donates the pointsx1,y1 andx2,y2
m donates the slope.
Plug the values to get
first tangent:
y-3 = -4/3(x-4) ==>3y-9 = -4x+16 simplify and get theanswer.
Similarly u can find second tangent as
y+3 = 4/3(x-4)

b) find the equations of the normal lines to this circle atthe same points . ( the normal line is perpendicular to the tangentline at that point)
I have already found the points. Now plug the values in the eqngiven below to get the eqn of normal (two normals)
the normal is given by:
y-Y = (-1/m)(x-X)

c) at what points do the two normal lines intersect?
The wo normals that you get are
3x-4y= 0.........................................(3)
and 3x+4y=0....................................(4)
solving eqn 3 and 4 u get x,y = 0,0 therefore the normalsmeet(intersect) at the origin.
Note: both the eqns of normal are independent of constant terms,which means they are passing through origin.

Let me know if u face any difficulty