The line `lx+my+n=0` is a normal to the parabola `y^2 = 4ax` if

Options

A. ` (( -n ) /(l ) , ( - 2am ) /( l)) `
B. ` ( ( -n ) /(l) , ( 2am )/( l) ) `
C. ` ( ( n ) / ( l) , ( - 2am )/ ( l)) `
D. ` (( n ) /( l) , ( 2am ) /( l)) `

Answer

C

Explanation

Tangent of the parabola ` y^ 2 = 4ax ` at point ` P ( x_ 1, y _ 1 ) ` is
` yy _ 1 = 2a ( x +x _ 1 ) `
`rArr 2a x _ 1 - yy _ 1 + 2ax = 0 " " ` … (i)
Which is also equation of the polar of the parabola `y ^ 2 = 4ax ` . Same as the line
` lx + my + n = 0 " " `... (ii)
On comparing both lines, we get
` ( 2a)/ (l ) = ( - y ) /( m) = ( 2ax ) /( n ) `
Taking first two parts
` ( y = - ( 2am ) / ( l)) `
Taking first and last parts,
` ( x = ( n ) / ( l) ) `
` therefore ` Required pole of the line is ` { ( n ) / (l) , ( - 2am ) /(l) } `.