1. ## im confused

helloo i need to get an answer for this question

Show that the line y = mx + c touches the circle x^2 + y^2 = a^2 if c^2 = a^2(1+m^2)

it's very hard for me i need some help plz.

2. Show that the line y = mx + c touches the circle x^2 + y^2 = a^2 if c^2 = a^2(1+m^2)
$x^2 + y^2 = a^2$
$x^2 + (mx + c)^2 = a^2$
$x^2 + m^2 x^2 + c^2+2mcx = a^2$
$(1 + m^2) x^2 +2mcx + (c^2- a^2)=0$

if the line y = mx + c touches the circle(or tangent to the circle) then
$discriminant = 0$
$d= (2mc)^2 -4(1 + m^2)(c^2- a^2)=0$
$4a^2-4c^2+4m^2a^2=0$
$c^2 = a^2+a^2m^2$
$c^2 = a^2(1+m^2)$

The point (p,q) lies on the circle x^2 + y^2 = a^2.
Show that the equation of the tangent to the circle at (p,q) is px + qy = a^2

4. the equation of the tangent to the circle having centre at origin is given by
$x_{0}x+y_{0}y=r^2$
$where (x_{0},y_{0}) are\ point\ of\ intersection$
$since\ (p,q)\ lies\ on\ the\ circle\ x^2 + y^2 = a^2.$
$therefore\ p^2 + q^2 = a^2 =(radius)^2$
therefore equation of the tangent
$px+qy= p^2 + q^2$