The normal at the point (b t₁², 2 b t₁) on a parabola meets the parabola again in the point (b t₂², 2 b t₂), then [AIEEE 2003]

Correct Answer is : t2=−t1−2t1t₂=-t₁-2/t₁t2=−t1−t12

Correct Answer is : t2=−t1−2t1t₂=-t₁-2/t₁t2=−t1−t12

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Parabola Part 1

Section: Mathematics

Question : 99 of 100

Marks: +1, -0

The normal at the point

(b t_1^2, 2 b t_1)

(b t_2^2, 2 b t_2)

Solution:

Equation of the normal to a parabola

y^2=4 b x

at point

(b t_1^2, 2 b t_1)

y=-t_1 x+2 b t_1+b t_1^3

As given, it also passes through

(b t_2^2, 2 b t_2)

then

2 b t_2=t_1 b t_2^2+2 b t_1+b t_1^3

2 t_2-2 t_1=-t_1 (t_2^2-t_1^2)

=-t_1 (t_2+t_1) (t_2-t_1)

\Rightarrow 2=-t_1 (t_2+t_1)

\Rightarrow t_2+t_1=-\frac{2}{t_1}

\Rightarrow t_2=-t_1-\frac{2}{t_1}

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