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 (Thinking) i need some help plz.

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- Oct 8th 2009, 08:47 AMMathboyim 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 (Thinking) i need some help plz. - Oct 8th 2009, 10:14 AMramiee2010Quote:

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

$\displaystyle x^2 + (mx + c)^2 = a^2$

$\displaystyle x^2 + m^2 x^2 + c^2+2mcx = a^2$

$\displaystyle (1 + m^2) x^2 +2mcx + (c^2- a^2)=0$

if the line y = mx + c touches the circle(or tangent to the circle) then

$\displaystyle discriminant = 0 $

$\displaystyle d= (2mc)^2 -4(1 + m^2)(c^2- a^2)=0$

$\displaystyle 4a^2-4c^2+4m^2a^2=0$

$\displaystyle c^2 = a^2+a^2m^2$

$\displaystyle c^2 = a^2(1+m^2)$ - Oct 8th 2009, 10:24 AMMathboy
Any help about this also?

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 - Oct 8th 2009, 10:55 AMramiee2010
the equation of the tangent to the circle having centre at origin is given by

$\displaystyle x_{0}x+y_{0}y=r^2$

$\displaystyle where (x_{0},y_{0}) are\ point\ of\ intersection$

$\displaystyle since\ (p,q)\ lies\ on\ the\ circle\ x^2 + y^2 = a^2.$

$\displaystyle therefore\ p^2 + q^2 = a^2 =(radius)^2$

therefore equation of the tangent

$\displaystyle px+qy= p^2 + q^2 $