Name: Pair of Tangent, Chord of Contact
Uploaded: 2019-07-19
Duration: 19 min 37 s
Description: This video contains all the details of the chord of the contact pair of tangent and cord bisected at a point. This concept is very useful for the students.

Question

Find the locus of a point $O$ when the three normals drawn from it are such that, the line joining the feet of two of them is always in a given direction for parabola $y^{2} = 4 a x .$

Accepted Answer

Let $A (a m_{1}^{2}, - 2 a m_{1})$ and $B (a m_{2}^{2}, - 2 a m_{2})$ are the coordinates of the feet of any two of the three normals.

The equation of normal to the parabola in slope form is $y = m x - 2 a m - a m^{3}$ if this passes through $O (h, k)$ then

$a m^{3} + m (2 a - h) + k = 0$

Now, the slope of the line joining $A B$ is given as,

$\Rightarrow m_{A B} = \frac{- 2 a (m_{1} - m_{2})}{a (m_{1}^{2} - m_{2}^{2})} (m_{1} \neq m_{2})$

$= \frac{- 2}{m_{1} + m_{2}}$ (which is constant)

Let $m_{1} + m_{2} = λ$

If $m_{1,} m_{2}, m_{3}$ are roots for cubic equation $a m^{3} + m (2 a - h) + k = 0$

This is the equation which represents three normals drawn from $(h, k)$

$m_{1} + m_{2} + m_{3} = 0$

$\Rightarrow m_{3} = - (m_{1} + m_{2}) = - λ$

$m_{3} m_{2} + m_{1} m_{3} + m_{1} m_{2} = \frac{2 a - h}{a}$

$\Rightarrow$ $m_{3} (m_{1} + m_{2}) + m_{1} m_{2} = \frac{2 a - h}{a}$

$m_{1} m_{2} m_{3} = \frac{- k}{a}$

$m_{3} (λ) - \frac{k}{a m_{3}} = \frac{2 a - h}{a}$

$\Rightarrow - λ (λ) + \frac{k}{a λ} = \frac{2 a - h}{a}$

$\Rightarrow - λ^{2} + \frac{k}{a λ} = \frac{2 a - h}{a}$

$\Rightarrow \frac{- a λ^{3} + k}{a λ} = \frac{2 a - h}{a}$

$\Rightarrow - a λ^{3} + k = 2 a λ - h λ$

$\Rightarrow a λ^{3} + (2 a - h) λ - k = 0$

Generalizing the locus of $(h, k)$ , we get,

$a λ^{3} + (2 a - x) λ = y \Rightarrow y = - λ x - 2 a (- λ) - a (- λ)^{3} w h e r e m_{3} = - λ$

This is a normal at $(a λ^{2}, - a λ)$ to the parabola $y^{2} = 4 a x$ .

Find the locus of a point $O$ when the three normals drawn from it are such that, the line joining the feet of two of them is always in a given direction for parabola $y^{2} = 4 a x .$

Important Questions on The Parabola (Continued)

Learn and Prepare for any exam you want !

Find the locus of a point O when the three normals drawn from it are such that, the line joining the feet of two of them is always in a given direction for parabola y2 = 4ax.

Important Questions on The Parabola (Continued)

Learn and Prepare for any exam you want !

Find the locus of a point $O$ when the three normals drawn from it are such that, the line joining the feet of two of them is always in a given direction for parabola $y^{2} = 4 a x .$