If $r_{1}$ and $r_{2}$ be the lengths of radii vectores of the parabola which are drawn at right angles to one another from the vertex, prove that $r_{1}^{\frac{4}{3}} r_{2}^{\frac{4}{3}} = 16 a^{2} (r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}) .$

Open in App

Solution

Let $r_{1}$ is inclined at an angle $θ$ with the axis then coordinates of point are $P (r_{1} cos θ, r_{1} sin θ)$

Then $r_{2}$ is inclined at $90 + θ$ with the axis, then coordinates of $Q$ are $(r_{2} sin θ, - r_{2} cos θ)$

Now $P$ lies on the parabola

${{(r}_{1} sin θ)}^{2} = 4 a r_{1} cos θ r_{1} {sin}^{2} θ = 4 a cos θ . . . . . . (i)$

$Q$ also lies on the parabola

${{(- r}_{2} cos θ)}^{2} = 4 a r_{2} sin θ r_{2} {cos}^{2} θ = 4 a sin θ . . . . . . . (i i)$

Dividing $(i)$ by $(i i)$

${tan}^{3} θ = \frac{r_{2}}{r_{1}} tan θ = {(\frac{r_{2}}{r_{1}})}^{\frac{1}{3}}$

$cos θ = \frac{1}{\sqrt{1 + {tan}^{2} θ}} \Rightarrow cos θ = \frac{r_{1}^{\frac{1}{3}}}{\sqrt{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}}} sin θ = tan θ cos θ \Rightarrow sin θ = \frac{r_{2}^{\frac{1}{3}}}{\sqrt{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}}}$

Substituting $sin θ$ and $cos θ$ in $(i)$

$r_{1} {⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{r_{2}^{\frac{1}{3}}}{\sqrt{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠}^{2} = 4 a \frac{r_{1}^{\frac{1}{3}}}{\sqrt{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}}} \frac{r_{1} r_{2}^{\frac{2}{3}}}{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}} = 4 a \frac{r_{1}^{\frac{1}{3}}}{\sqrt{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}}} r_{1}^{\frac{2}{3}} r_{2}^{\frac{2}{3}} = 4 a \sqrt{r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}}}$

Squaring both sides

$r_{1}^{\frac{4}{3}} r_{2}^{\frac{4}{3}} = 16 a^{2} (r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}})$

Suggest Corrections

Similar questions

y^{2} = 4 a x

r_{1}

r_{2}

\frac{r_{1}^{4 / 3} r_{2}^{4 / 3}}{r_{1}^{2 / 3} + r_{2}^{2 / 3}}

r_{1}

r_{2}

y^{2} = 4 a (x - 8 a) .

r_{1}

r_{2}

r

r = r_{1} + r_{2}

{r_{1}}^{2} + {r_{2}}^{2} = r^{2}

r_{1} + r_{2} < r

{r_{1}}^{2} + {r_{2}}^{2} < r^{2}

Join BYJU'S Learning Program