Question

Find the locus of the poles of the normal chords of the parabola y^2 = 12 x.
​

Answer 1

Answer:

Let the pole of the parabola y

2

=4ax be P(h,k)

Equation of normal chord is

y=−tx+2at+at

3

y+tx−2at−at

3

=0 ......(i)

Equation of the polar with respect to P is T=0

ky=2a(x+h)

2ax−ky+2ah=0 .......(ii)

Now (i) and (ii) represents the equation of same line, so comparing both equations

2a

t

=

−k

1

=

2ah

−2at−at

3

⇒

2a

t

=

−k

1

=

2h

−2t−t

3

⇒

2a

t

=

−k

1

⇒t=

k

−2a

.......(iii)

Also

2a

t

=

2h

−2t−t

3

⇒

a

1

=

h

−2−t

2

⇒

a

h

+2=−t

2

Substituting t from (iii)

a

h+2a

=−(

k

−2a

)

2

⇒k

2

(h+2a)=−4a

3

⇒k

2

(h+2a)+4a

3

=0

Generalizing the equation, we get

y

2

(x+2a)+4a

3

=0

So, option B is correct.

Answered By

Step-by-step explanation:

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