The lie x-b- y-0 cuts the parabola y2-4ax at Pt1 and Qt2- If b-2a-4a- and - then the values -t1t2- is-arewhere - repreasents the greatest integer function and t1-t2 are parametic points

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Focal Chord

The lie x-b+λ...

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 option is 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

Q. The lie

x - b + λ y = 0

cuts the parabola

y^{2} = 4 a x

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)

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.

Q. If the end points

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and

Q (t_{2})

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y^{2} = 4 a x

satisfy the relation

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

then prove that the chord always through a fixed point. Find that point also?

If the chord joining the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ of the parabola $y^{2} = 4 a x$ passes through the focus of the parabola, then

Q. The tangents to the parabola

y^{2} = 4 a x

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Focal Chord

Standard XIII Mathematics

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The lie x−b+λy=0 cuts the parabola y2=4ax at P(t1) and Q(t2). If b∈[2a,4a] and λ∈R, then the value(s) [t1t2] is/are (where [.] repreasents the greatest integer function and t1,t2 are parametic points)

The correct option is D −2Given P(t1)≡(at21,2at1) and Q(t2)≡(at22,2at2) Now line x−b+λy=0 always passes through (b,0) Slope of PR= Slope of RQ ⇒2at1at21−b=2at2at22−b⇒t1t2=−ba Now from b∈[2a,4a] t1t2∈[−4,−2]

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)