Normal to a Parabola

Question

Normals are drwan at three different points on parabola !!y^2=4x!! pass through !!(h,k)!!. Then

(a) !!h> 0!!

(b) !!h> 1!!

(c) !!h> 2!!

(d) !!h> 3!!

Solution

Equation of the normal on parabola !!y^2=4ax!!,

!!y= mx- 2am-m^3!!

According to question !!a=1!! and normal passing through the point !!(h,k)!!

!! k= mh-2m-m^3!!

!!=> m^3+(2-h)m+k=0!!

Suppose !!m_1,m_2,m_3!! are roots of the equation.

!!m_1+m_2+m_3=0 ......(1)!!

!!m_1m_2+m_2m_3+m_3m_1= 2-h ......(2)!!

!!m_1m_2m_3= -k!!

As !!m_1,m_2,m_3!! are real

!!m_1^2+m_2^2+m_3^2>0!!

!!(m_1+m_2+m_3)^2-2(m_1m_2+m_2m_3+m_3m_1)>0!!

From the equation !!(1)!! and !!(2)!!, we get

!!0-2(2-h)>0!!

!!=> -4+2h>0!!

!!=>h> 2!!

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