The equation of chord joining 2 point P - and Q - in a hyperbola x2- a 2- y 2- b 2-1 is x - a -cos-2- y - b -sin-2-cos-2-A- TrueB- False

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Position of a Point W.R.T Hyperbola

The equation ...

Question

The equation of chord joining 2 point P $(α)$ and Q $(β)$ in a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 i s \frac{x}{a} . c o s (\frac{α - β}{2}) - \frac{y}{b} . s i n (\frac{α + β}{2}) = c o s [\frac{α + β}{2}]$ .

True

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False

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Solution

The correct option is A
True

The points joined by the chord are P $a s e c α, b t a n α) a n d Q (a s e c β, b t a n β)$

$∴$ equation of chord is , $y - b t a n α = \frac{b . (t a n α - t a n β)}{a (s e c α - s e c β)} . (x - a s e c α)$

$a y s e c α - a y s e c β - a b s e c α . t a n α + a b . s e c α . t a n β$

$a y s e c α - a y s e c β - a b s e c β . t a n α$

= $b x . t a n α - b x . t a n β + a b s e c α t a n β$

Multiplying throughout by $\frac{c o s α . c o s β}{a b}$

$\frac{y}{b} . c o s β - \frac{y}{b} . c o s α + \frac{x}{a} . c o s α . s i n β - \frac{x}{a} . c o s β . s i n α$

= $s i n β - s i n α$

$\frac{x}{a} . s i n (β - α) + \frac{y}{b} (-) 2. s i n (\frac{α + β}{2}) . s i n (\frac{α - β}{2})$

= $2. c o s \frac{α + β}{2} . s i n [\frac{β - α}{2}]$

$\frac{x}{a} .2 s i n (\frac{β - α}{2}) c o s (\frac{β - α}{2}) - \frac{y}{b} .2 s i n (\frac{α + β}{2}) . s i n (\frac{α - β}{2})$

= $2 c o s \frac{α + β}{2} . s i n (\frac{β - α}{2})$

Dividing by $2 s i n (\frac{β - α}{2})$

$\frac{x}{a} . c o s (\frac{β - α}{2}) - \frac{y}{b} . s i n \frac{α + β}{2} = c o s \frac{α + β}{2}$

Hence the given statement is true

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Similar questions

Q. If

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then the chord joining the points

α

and

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\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

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Q. If

α

and

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\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

and the chord joining these two points passes
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c o s \frac{α - β}{2}

The locus of a point $P (α, β)$ moving under the condition that the line $y = α x + β$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$

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\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

Q. If the chord joining the points

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The equation of chord joining 2 point P(α) and Q(β) in a hyperbola x2a2 − y2b2= 1 is xa.cos (α−β2) − yb.sin (α+β2) = cos [α+β2].

The equation of chord joining 2 point P $(α)$ and Q $(β)$ in a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 i s \frac{x}{a} . c o s (\frac{α - β}{2}) - \frac{y}{b} . s i n (\frac{α + β}{2}) = c o s [\frac{α + β}{2}]$ .