"For a given parabola and a given point (h, k), this cubic in m has three roots say m1, m2, - Maths - Conic Sections

Question

"For a given parabola and a given point (h, k), this cubic in m has
three roots say m1, m2, m3 i e from (h, k) three normals can be
drawn to the parabola whose slopes are m1, m2, m3 For this cubic,
we have m1 + m2 + m3 = 0,
m1 m2 + m 2
m3 + m3 m1 = (2a – h)/a and
m1 m2 m3 = –k/a "
Prove :

1 ​m1 m2 + m2 m3 + m3 m1 = (2a – h)/a

​2 ​m1 m2 m3 = –k/a

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

We know that equation of normal of m-slope of parabola is y=mx-2am-am3Normal passes through h,kk=mh-2am-am3⇒am3+m2a-h+k=0 Roots will be m1,m2,m3Hencem1+m2+m3=-Coeffcient of m2Coefficient of m3=0a=0m1m2+m2m3+m3m1=Coeffcient of mCoefficient of m3=2a-ham1m2m3=-Constant termCoefficient of m3=-ka

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