Question

If x+y=k is a normal to y2=12x, then k= ​

Answer 1

Comparing y2=12x with standard equation of parabola y2=4ax we have a=3

y=mx+c is a normal to parabola if C=−2am−am3

Comparing x+y=k with y=mx+c we get,

m=−1c=k

k=−2a(−1)−a(−1)3

k=−6(−1)−3(−1)

=6+3

=9

Hence B is the correct answer.

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Answer 2

$\large\underline\mathrm\red{Given:-}$

• if x + y = k is a normal to y² = 12x

• Comparing y² = 12x with standard equation of parabola y² = 4ax

$\implies$4a = 12

$\large\underline\mathrm\red{Then}$, a = 3

y = mx + c is normaly to parabola if c = -2am - am³

$\large\underline\mathrm\red{And}$ x + y = k

$\large\underline\mathrm\red{Then}$, y = - x + k

Comparing above equation with y = mx +c,

$\large\underline\mathrm\red{We \: gets,}$

$\implies$m = -1 and c = k

$\implies$k = -2am - am³ $\large\underline\mathrm\red{(k = c)}$

$\large\underline\mathrm\red{put \: a \: and \: m \: is \: above \: equation,}$

$\implies$k = -2 × 3 × (-1) -3 × (-1)³

$\implies$k = (6 + 3)

$\implies$k = 9

$\large\underline\mathrm\red{so, \: the \: value \: of \: k \: is \: 9.}$

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