Auxiliary circle - class-XII
Description: auxiliary circle | |
Number of Questions: 42 | |
Created by: Anumati Koshy | |
Tags: two dimensional analytical geometry-ii hyperbola mathematics and statistics conic sections maths |
For the hyperbola $\dfrac{x^2}{64}-\dfrac{y^2}{36}=1$, the equation of director circle is
The equation of auxillary circle of hyperbola is $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
The equation of director circle of $\dfrac{x^2}{64}-\dfrac{y^2}{49}=1$ is
The length of diameter of director circle of hyperbola $\dfrac{x^2}{49}-\dfrac{y^2}{25}=1$, is
The equation of director circle for $\dfrac{x^2}{100}-\dfrac{y^2}{36}=1$, is
The equation of director circle of hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is
The circle passing through the vertices of hyperbola is called
The intersection point of,a perpendicular on tangent of a hyperbola from the focus and a tangent lies on
If $\theta$ is eliminated from the equations $a\sec\theta - x\tan\theta = y \mbox{ and } b\sec\theta + y\tan\theta = x$ ($a$ and $b$ are constant), then the eliminant denotes the equation of
If pair of tangents are drawn from any point $(p)$ on the circle ${x^2} + {y^2} = 1$ to the hyperbola $\frac{{{x^2}}}{2} - \frac{{{y^2}}}{1} = 1$ such that locus of circumcenter of triangle formed by pair of tangents and chord of contact is ${\lambda _1}{x^2} - 2{\lambda _2}{y^2} = 2{\left( {\frac{{{x^2}}}{2} - {y^2}} \right)^2}$, then
Find the range of $p$ such that no perpendicular tangents can be drawn to the hyperbola $\dfrac{x^2}{(-p^2 + 6p + 5)} - \dfrac{y^2}{(-p - 3)} = 1$, i.e. the director circle of the given hyperbola is imaginary.
For the hyperbola $\dfrac{x^2}{49}-\dfrac{y^2}{25}=1$, the equation of auxillary circle is
The radius of the director circle of the ellipse $9{x^2} + 25{y^2} - 18x - 100y - 116 = 0$ is
The equation of auxillary circle of $\dfrac{x^2}{64}-\dfrac{y^2}{36}=1$ is
For the hyperbola $\dfrac{x^2}{15}-\dfrac{y^2}{10}=1$, the equation of auxillary circle is
If ${e _1}$and ${e _2}$ are the eccentricities of the hyperbolas $xy = 9$ and ${x^2} - {y^2} = 25$ ,then( ${e _1}$,${e _2}$) lie on a circle ${C _1}$with centre origin then the ${(radius)^2}$ of the director circle of ${C _1}$is
The equation of auxillary circle is $\dfrac{x^2}{25}-\dfrac{y^2}{16}=1$
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 :
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 -
The radius of the director circle of the hyperbola $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ is
If one of the directrix of hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{b}=1$ is $x=-\dfrac{9}{5}$. Then the corresponding focus of hyperbola is?
The equation of the director circle of the hyperbola $\dfrac{x^2}{81}- \dfrac{y^2}{16}=1$ is
The equation of the director circle of the hyperbola $\dfrac{x^2}{36}- \dfrac{y^2}{16}=1$ is
Auxiliary circle of a hyperbola is defined as:
The circle with major axis as diameter is called the auxiliary circle of the hyperbola.
If $a>b,$ then the equation of auxiliary circle is
The equation of director circle of the hyperbola $-\dfrac{x^2}{a^2}+ \dfrac{y^2}{b^2}=1$, if $b>a$, is
The radius of director circle of the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}{9}=1$ is
The equation of director circle of $-\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, If $b<a$ is:
The equation of the auxiliary circle of the hyperbola $4x^2-9y^2=36$ is
If any tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ with centre $C$, meets its director circle in $P$ and $Q$, then:
The radius of director circle of hyperbola is $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
The equation of director circle of $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is:
The director circle intersects its hyperbola in _______ number of points.
The radius of the director circle of the hyperbola $\dfrac{x^2}{a(a+4b)}-\dfrac{y^2}{b(2a-b)}=1; 2a > b > 0$ is:
The diametre of director circle of hyperbola $\dfrac{x^2}{25}-\dfrac{y^2}{16}=1$
The equation of director circle of hyperbola is $\dfrac{x^2}{36}-\dfrac{y^2}{25}=1$ is
Point P is on the orthogonal hyperbola $x^2 - y^2 = a^2$. Point P' is the perpendicular projection of P on the x-axis. Then, $|PP'|^2$ is equal to the power of point P' relative to which circle?
The pole of the line $lx + my + n = 0$ with respect to the hyperbola $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, is
The number of points from where a pair of perpendicular tangents can be drawn to the hyperbola, $ x^2 \sec^2\alpha-y^2 \cos ec^2\alpha=1, \alpha\in(0,\dfrac{\pi}4) $ are
The locus of the point of intersection of two perpendicular tangents to the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is
If the tangent at the point $(h, k)$ to the hyperbola $\dfrac{x^2}{a^2}\, -\, \dfrac{y^2}{b^2}\, =\, 1$ cuts the auxiliary circle in points whose ordinates are $y _1$ and $ y _2$, then $\dfrac{1}{y _1} + \dfrac{1}{y _2} =$.
Find the range of $p$ such that a unique pair of perpendicular tangents can be drawn to the hyperbola $\dfrac{x^2}{(p^2 - 4)} - \dfrac{y^2}{(p^2 + 4p + 3)} = 1$, i.e. the director circle of the given hyperbola is a point.