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

Options

A. `mn = al ^ 2 `
B. ` lm = an ^ 2 `
C. `ln = am ^ 2 `
D. None of the above

Answer

C

Explanation

Given ,parabola, ` y ^ 2 = 4 ax `
` rArr 2y ( dy ) /( dx ) = 4a `
` rArr ( d y ) /( dx ) = (2a ) /( y ) " " ` … (i)
which is the slope of tangent
Given ` lx + my + n = 0 `
is an equation of tangent of the parabola
`y^ 2 =4a x `
` therefore ` Slope of tangent ` = - ( 1 ) /( m ) " "` ... (ii)
From Eqs (i) and (ii)
` (2a )/( y ) = - ( l ) / ( m ) rArr y = ( - 2am ) /( l ) `
` because y ^ 2 = 4ax `
` rArr ( 4a ^ 2 m^ 2 ) /( l ^ 2 ) = 4ax`
` rArr x = ( am^ 2 ) /( l^ 2 ) `
On putting the values of x and y in the following equation
` l ( ( am^2 ) /( l^2 )) +m ( ( -2 am ) /( l ) ) + n = 0`
` ( am ^ 2 ) /( l ) -(2am ^ 2 ) /( l) + n = 0 `
` rArr ( am^ 2 ) / ( l) = n rArr am ^ 2 = n l`
Which is the required relation.