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  • The equations of the normal at the ends of the latus rectum of the parabola  <img alt="\mathrm{y^2=4 a x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7By%5E2%3D4%20a%2

The equations of the normal at the ends of the latus rectum of the parabola  \mathrm{y^2=4 a x} are given by

Option: 1

\mathrm{x^2-y^2-6 a x+9 a^2=0}


Option: 2

\mathrm{x^2-y^2-6 a x-6 a y+9 a^2=0}


Option: 3

\mathrm{x^2-y^2-6 a y+9 a^2=0}


Option: 4

\mathrm{\text { None of these }}


Answers (1)

best_answer

The coordinates of the ends of the latus rectum of the parabola \mathrm{y^2=4 a x \, \, are \, \, (a, 2 a) \, \, and \, \, (a,-2 a)} respectively.

The equation of the normal at \mathrm{(a, 2 a) \text { to } y^2=4 a x_{\text {\, \, is }} y-2 a=\frac{-2 a}{2 a}(x-a)\left\{\text { using } y-y_1=\frac{-y_1}{2 a}\left(x-x_1\right)\right\}}

Or    \mathrm{x+y-3 a=0}                \mathrm{.....(i)}

Similarly the equation of the normal at \mathrm{(a,-2 a) \text { is } x-y-3 a=0 \quad \ldots . \text { (ii) }}

The combined equation of (i) and (ii) is  \mathrm{x^2-y^2-6 a x+9 a^2=0}

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