Hey, i'm being asked to find the equation of the curve that is formed by the tangents of the collection of lines from the following parametric equation: "(m^2+5m+4)x+(4-m^2)y+9m=0 i learned that i can find it by:
1-Rearrange the equation in terms of m: (x-y)m^2+(5x+9)m+(4x+4y)=0
2-With that quadratic equation i can find the equation of the curve by using the discriminant, so now i have (5x+9)^2-4(x-y)(4x+4y)=0
3-Expanding and rearranging the equation i finally have: 9x^2+16y^2+90x+81=0 which is the equation of an elipse.
But i want to know why i'm using the discriminant to find that curve, i have an exam about this in 2 months and i don't want to just memorize how to do it.
Also i have another question about this problem: does the point (-9/5,-9/5) belongs to the collection of lines? i assumed that it didn't because substituting in the original equation of the collection of lines (m^2+5m+4)x+(4-m^2)y+9m=0) i had "-72/5=0" as a result, but the point is a solution to the elipse "9x^2+16y^2+90x+81=0" so it has to be a solution to one of the tangent lines to the elipse what am i doing wrong here?
Sorry for my bad english, not my native language.
1-Rearrange the equation in terms of m: (x-y)m^2+(5x+9)m+(4x+4y)=0
2-With that quadratic equation i can find the equation of the curve by using the discriminant, so now i have (5x+9)^2-4(x-y)(4x+4y)=0
3-Expanding and rearranging the equation i finally have: 9x^2+16y^2+90x+81=0 which is the equation of an elipse.
But i want to know why i'm using the discriminant to find that curve, i have an exam about this in 2 months and i don't want to just memorize how to do it.
Also i have another question about this problem: does the point (-9/5,-9/5) belongs to the collection of lines? i assumed that it didn't because substituting in the original equation of the collection of lines (m^2+5m+4)x+(4-m^2)y+9m=0) i had "-72/5=0" as a result, but the point is a solution to the elipse "9x^2+16y^2+90x+81=0" so it has to be a solution to one of the tangent lines to the elipse what am i doing wrong here?
Sorry for my bad english, not my native language.