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Focal Chord

The lie x - b...

Question

The lie $x - b + λ y = 0$ cuts the parabola $y^{2} = 4 a x$ at $P (t_{1})$ and $Q (t_{2})$ . If $b \in [2 a, 4 a]$ and $λ \in R$ , then the value(s) $[t_{1} t_{2}]$ is/are
(where $[.]$ repreasents the greatest integer function and $t_{1}, t_{2}$ are parametic points)

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Solution

The correct options are
B $- 4$
C $- 3$
D $- 2$
Given $P (t_{1}) \equiv (a t_{1}^{2}, 2 a t_{1})$ and $Q (t_{2}) \equiv (a t_{2}^{2}, 2 a t_{2})$
Now line $x - b + λ y = 0$ always passes through $(b, 0)$
Slope of $P R =$ Slope of $R Q$
$\Rightarrow \frac{2 a t_{1}}{a t_{1}^{2} - b} = \frac{2 a t_{2}}{a t_{2}^{2} - b} \Rightarrow t_{1} t_{2} = - \frac{b}{a}$
Now from $b \in [2 a, 4 a]$
$t_{1} t_{2} \in [- 4, - 2]$

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Similar questions

x - b + λ y = 0

y^{2} = 4 a x

P (t_{1})

Q (t_{2})

b \in [2 a, 4 a]

λ \in R

[t_{1} t_{2}]

[.]

t_{1}, t_{2}

If the tangents at P(t1) and Q(t2) in the parabola $y^{2} = 4 a x$ meet in T, then find the coordinates of point T.

y^{2} = 4 a x

P (t_{1})

Q (t_{2})

R

△ P Q R

P (t_{1})

Q (t_{2})

y^{2} = 4 a x

t_{1} t_{2} = k (c o n s t a n t)

P (t_{1}), Q (t_{2})

y^{2} = 8 x .

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