If circle x^2 + y^2 = a^2 intersects hyperbola xy = c^2 in four points P(x₁, y₁), Q(x₂,

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10-2. Parabola, Ellipse, Hyperbola

normal

If the circle $x^2 + y^2 = a^2$ intersects the hyperbola $xy = c^2 $ in four points $ P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4), $ then

$x_1 + x_2 + x_3 + x_4 = 0$

$y_1 + y_2 + y_3 + y_4 = 0$

$x_1 x_2 x_3 x_4 = c^4$

all of the above

Solution

solving $xy = c^2$ and $x^2 + y^2 = a^2$
$x^2 +\frac{{{c^4}}}{{{x^2}}}\, = a^2$
$x^4- ax^3- a^2x^2 + ax + c^4 = 0$
$\Rightarrow$ ${\sum x _i}\, = \,0$;${\sum y _i}\, = \,0$
$x_1 x_2 x_3 x_4 = c^4$ $\Rightarrow$ $y_1 y_2 y_3 y_4 = c^4$

Standard 11

Mathematics

Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola $16y^2 – 9x^2 = 1 $ is

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The foci of a hyperbola are $( \pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2 x+3 y=6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$ – and $y$-axes are $a$ and $b$ respectively, then $|6 a|+|5 b|$ is equal to $……….$.

hard

(JEE MAIN-2023)

The normal to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1$ at the point $(8,3 \sqrt{3})$ on it passes through the point

medium

(JEE MAIN-2022)

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medium

If $ P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3) $ and $ S(x_4, y_4) $ are $4 $ concyclic points on the rectangular hyperbola $x y = c^2$ , the co-ordinates of the orthocentre of the triangle $ PQR$ are :

normal

If the circle $x^2 + y^2 = a^2$ intersects the hyperbola $xy = c^2 $ in four points $ P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4), $ then

Solution

Similar Questions

Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola $16y^2 – 9x^2 = 1 $ is

The normal to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1$ at the point $(8,3 \sqrt{3})$ on it passes through the point

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$

If $ P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3) $ and $ S(x_4, y_4) $ are $4 $ concyclic points on the rectangular hyperbola $x y = c^2$ , the co-ordinates of the orthocentre of the triangle $ PQR$ are :

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