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Standard XII

Mathematics

Diameter : Hyperbola

If the radius...

Question

If the radius of the auxiliary circle of hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{4} = 1$ is $r$ and if the abscissa of the points with eccentric angle 60 degrees on the hyperbola and auxiliary circle respectively are $p$ and $q$ , then what is the value of $p + q - r$ ?

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Solution

Equation of auxiliary circle of hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is $x^{2} + y^{2} = a^{2}$ .
In this case, $a^{2} = 16$ .
Therefore, Required equation is $x^{2} + y^{2} = 16$ .
Corresponding points $N (4 cos 60, 4 sin 60)$ , $N (2, 2 \sqrt{3})$ on circle and
$P (4 sec 60, 2 tan 60)$ , $P (8, 2 \sqrt{3})$ on hyperbola.
Hence $r = 4, p = 8, q = 2$ . So, $p + q - r = 6$ .

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Similar questions

Q. If the tangent at the point

(p, q)

on the hyperbola

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

cuts the auxiliary circle in points whose ordinates are

y_{1}

and

y_{2}

then show that

q

is harmonic mean of

y_{1}

and

y_{2} .

Q. Let

P, Q, R

be three points on the ellipse

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

and

P^{'}, Q^{'}

and

R^{'}

be the corresponding points on the auxiliary circle. Then,
Area of

Δ

PQR : Area of

Δ

P' Q' R'

=

Q. If the chords of contact of tangents drawn from

P

to the hyperbola

x^{2} - y^{2} = a^{2}

and its auxiliary circle are at right angle, then

P

lies on :

P

is a point on the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

and

Q

is its corresponding point on the auxiliary circle. lf the locus of the point of intersection of the normal at

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to the respective curves is circle, then its radius is:

Let $P, Q, R$ be three points on the ellipse $x^{2} + 4 y^{2} = 4$ and $P^{'}, Q^{'}, R^{'}$ be the corresponding points on the auxiliary circle. Then Area of $△ P^{'} Q^{'} R^{'}$ $:$ Area of $△ P Q R$ is

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If the radius of the auxiliary circle of hyperbola x216−y24=1 is r and if the abscissa of the points with eccentric angle 60 degrees on the hyperbola and auxiliary circle respectively are p and q, then what is the value of p+q−r ?

Equation of auxiliary circle of hyperbola x2a2−y2b2=1 is x2+y2=a2.In this case, a2=16.Therefore, Required equation is x2+y2=16.Corresponding points N(4cos60,4sin60), N(2,2√3) on circle andP(4sec60,2tan60) , P(8,2√3) on hyperbola.Hence r=4,p=8,q=2. So, p+q−r=6.

If the radius of the auxiliary circle of hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{4} = 1$ is $r$ and if the abscissa of the points with eccentric angle 60 degrees on the hyperbola and auxiliary circle respectively are $p$ and $q$ , then what is the value of $p + q - r$ ?