You visited us 1 times! Enjoying our articles? Unlock Full Access!

Chord of Contact

Three normals...

Question

Three normals drown from any point to the parabola $y^{2} = 4 a x$ cut the line $x = 2 a$ in points whose ordinates are in arithmetic progression . If the slopes of the normals be $m_{1}, m_{2}$ and $m_{3}$ then ( $\frac{m_{1}}{m_{2}}$ ) ( $\frac{m_{3}}{m_{2}}$ ) is equal to

Open in App

Solution

Equation of normal $y + t x = 2 a t + a t^{3}$ .......(I)

$\Rightarrow$ Point of intersection with $x = 2 a$

$\Rightarrow y + 2 a t = 2 a t + a t^{3}$

$\Rightarrow y = a t^{3}$

Let the ordinates be $(a t_{1}^{3}, a t_{2}^{3}, a t_{3}^{3})$

Given:

$2 a t_{2}^{3} = a t_{1}^{3} + a t_{3}^{3}$

$2 t_{2}^{3} = t_{1}^{3} + t_{3}^{3}$

from (I) $t_{1} + t_{2} + t_{3} = 0$

$\Rightarrow t_{1} + t_{3} = - t_{2}$

$2 t_{2}^{3} = (t_{1} + t_{3})^{3} - 3 t_{1} t_{3} (t_{1} + t_{3})$

$2 t_{2}^{3} = (- t_{2})^{3} - 3 t_{1} t_{3} (- t_{2})$

$3 t_{2}^{3} = 3 t_{1} t_{2} t_{3}$

$\Rightarrow t_{2}^{2} = t_{1} t_{3}$

since $m = - t$

$m_{2}^{2} = m_{1} m_{3}$

$\Rightarrow (\frac{m_{1}}{m_{2}}) (\frac{m_{3}}{m_{2}}) = 1$

Hence the answer is $1$ .

Suggest Corrections

Similar questions

m_{1}, m_{2}

m_{3}

P

y^{2} = 4 x .

m_{1} m_{2} = α

P

α

P

m_{1}, m_{2}, m_{3}

y^{2} = 4 x .

P

m_{1}, m_{2} = α

α

P

m_{1}

m_{2}

m_{1} \cdot m_{2} = α

P

y^{2} = 4 x

α

x = 2 a

P

m_{1} \cdot m_{2} = α

P

y^{2} = 4 x

α

Join BYJU'S Learning Program