Question

If the parabola y^2 = 4ax passes through the point (3, 2), then the length of its latus rectum is

Answer 1

$\textsf{Given parabola is }$

$\mathsf{y^2=4ax}$

$\textsf{It passes through (3,2)}$

$\implies\mathsf{2^2=4a(3)}$

$\implies\mathsf{4=4a(3)}$

$\implies\mathsf{1=3a}$

$\implies\mathsf{a=\frac{1}{3}}$

$\textsf{Length of the latusrectum=4a}$

$\textsf{Length of the latusrectum= 4(\frac{1}{3}) }$

$\implies\textsf{Length of the latusrectum= \frac{4}{3} }$

Answer 2

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Length\:of\:latus\:rectum=\frac{4}{3}\:units}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{ \underline \bold{given :}} \\ \tt{: \implies eqn \: of \: parabola \: ({y}^{2} =4 ax}) \\ \\ \tt{: \implies point \: on \: parabola = (3, 2) } \\ \\ \red{ \underline \bold{to \: find :}} \\ \tt{: \implies length \: of \: latus \: rectum = ?}$

• According to given question :

$\tt {: \implies {y}^{2} = 4ax} \\ \\ \to \text{3,2 \: is \: on \: parabola \: so, \: the}\\ \text{Eqn \: satisfy \: this \: point} \\ \\ \tt{: \implies { 2}^{2} = 4 \times a \times 3} \\ \\ \tt{: \implies 4 = 12a} \\ \\ \tt{: \implies a = \frac{4}{12} } \\ \\ \green{\tt{: \implies a = \frac{1}{3}}} \\ \\ \green{ \tt{: \implies Eqn \: of \: parabola \: is \: \: {y}^{2} = \frac{4}{3}x}} \\ \\ \bold{As \: we \: know \: that} \\ \tt{: \implies Length \: of \: latus \: rectum = 4a} \\ \\ \tt{: \implies Length \: of \: latus \: rectum = 4 \times \frac{1}{3}} \\ \\ \green{\tt{: \implies Length \: of \: latus \: rectum = \frac{4}{3} \: units}}$