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

Two equal parabolas have the same vertex and their axes are at right angles. Prove that the common tangent touches each at the end of a latus rectum.

Accepted Answer

Taking the axis of the first parabola as $x$ -axis and the axis of the other parabola as $y$ -axis, their equations can be written as

$y^{2} = 4 a x . . . (i)$

and $x^{2} = 4 a y . . . (i i)$

(As the parabolas are equal, their latus rectum will be equal, $4 a$ ).

Equation to any tangent to $(i)$ is $y = m x + \frac{a}{m} \dots . . (i i i)$

To solve $(i i)$ and $(i i i)$ , we put the value of $y$ from $(i i i)$ in $(i i)$ , we get

$x^{2} = 4 a (m x + \frac{a}{m})$

$m x^{2} - 4 a m^{2} x - 4 a^{2} = 0 \dots . . (i v)$

If equation $(i i i)$ touches $(i i)$ , then $(i v)$ must be a perfect square, i.e., the discriminant of $(i v)$ must be zero.

Hence. ${(- 4 a m^{2})}^{2} - 4 . m (- 4 a^{2}) = 0$

so, either $m = 0$ or $m^{3} = - 1$

$\Rightarrow m = - 1$

If $m = 0$ , line is $x$ -axis, which is not a tangent to $(i)$ .

So, putting $m = - 1$ in $(i i i)$ , the equation to the common tangent is

$y = - x - a$

$y + x + a = 0 \dots \dots (v)$

To get the point of contact, putting the value of $m$ in $(i v)$ ,

$- x^{2} - 4 a x - 4 a^{2} = 0$

${(x + 2 a)}^{2} = 0$

whence $x = - 2 a$ .

Putting this value in $(i i)$ , $y = a$ .

So $(v)$ touches $(i i)$ at $(- 2 a, a)$ which is one end of the latus rectum of the parabola $x^{2} = 4 a y$ .

Similarly, solving equation $(i)$ and $(v)$ , we can show that the point of contact is $(a, - 2 a)$ , which is the end of the latus rectum of the parabola $y^{2} = 4 a x$ .

Hence proved.

Two equal parabolas have the same vertex and their axes are at right angles. Prove that the common tangent touches each at the end of a latus rectum.

Important Questions on Conic Sections. The Parabola

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