Through the vertex O of the parabola y 2-4 ax two chords OP - OQ are drawn and the circles on OP -OQ as diameter intersect in R- If -1- -2- are the inclinations of the tangents at P - Q on the parabola and the line through O - R respectively- then the value of -1-2 isA- -2 tan-B- -2 tan -C- 0D- 2 -

The correct option is A Let P(t_1) amp; Q(t_2) ⇒ Slope of tangent at P(1/t_1) amp; at Q(1/t_2) ⇒ cot θ_1=t_1 and cot θ_2=t_2 Slope of PQ=2/t_1+t_2=tan ϕ ...

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Byju's Answer

Standard XI

Mathematics

Distinguish Acute Angle Bisectors and Obtuse Angle Bisectors

Through the v...

Question

Through the vertex O of the parabola $y^{2} = 4 a x$ two chords OP & OQ are drawn and the circles on OP &OQ as diameter intersect in R. If $θ_{1}$ , $θ_{2}$ & $ϕ$ are the inclinations of the tangents at P & Q on the parabola and the line through O, R respectively, then the value of $c o t θ_{1} + c o t θ_{2}$ is

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Solution

The correct option is A
Let $P (t_{1})$ & $Q (t_{2})$
$\Rightarrow$ Slope of tangent at $P (\frac{1}{t_{1}})$ & at $Q (\frac{1}{t_{2}})$ $\Rightarrow c o t θ_{1} = t_{1}$ and $c o t θ_{2} = t_{2}$
Slope of $P Q = \frac{2}{t_{1} + t_{2}} = t a n ϕ$
$\Rightarrow t a n ϕ = - \frac{1}{2} (c o t θ_{1} + c o t θ_{2})$ $\Rightarrow c o t θ_{1} + c o t θ_{2} = - 2 t a n θ$

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Similar questions

Q. Through the vertex

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Through the vertex O of the parabola y2=4ax two chords OP & OQ are drawn and the circles on OP &OQ as diameter intersect in R. If θ1,θ2 & ϕ are the inclinations of the tangents at P & Q on the parabola and the line through O, R respectively, then the value of cotθ1+cotθ2 is

The correct option is A Let P(t1) & Q(t2) ⇒ Slope of tangent at P(1t1) & at Q(1t2) ⇒cotθ1=t1 and cotθ2=t2 Slope of PQ=2t1+t2=tanϕ ⇒tanϕ=−12(cotθ1+cotθ2) ⇒cotθ1+cotθ2=−2tanθ