Hyperbola practice · Mixed difficulty

An ellipse intersects the hyperbola $2 x^2-2 y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

If $\frac{x^2}{\lambda+3}+\frac{y^2}{2-\lambda}=1$ represents a hyperbola, then

The locus of the point of intersection of the straight lines, $\mathrm{tx}-2 \mathrm{y}-3 \mathrm{t}=0$ $x-2 t y+3=0(t \in R)$, is :

The eccentricity of the hyperbola $5{x^2} - 4{y^2} + 20x + 8y = 4$ is

If circle ${x^2} + {y^2} + 2\mathrm{g}x + 2fy + k = 0$ intersect hyperbola $xy = {c^2}$ at four points $\left( {{x_i},{y_1}} \right),i = 1,2,3,4$ then

$P F \times PG=$

Let $S=\left\{(x, y) \in R: \frac{y^2}{1+r}-\frac{x^2}{1-r}=1\right\}$, where $r \neq \pm 1$. Then, $S$ represents

If tangent to the hyperbola $\frac{x^2}{4}-y^2=1$ cut the circle $x^2+y^2=16$ at two distinct point $A$ and $B$, then the locus of mid-point of $A$ and $B$ is

The tangent to the hyperbola $x y=c^2$ at point $P(t)$ intersects the $X$-axis at $T$ and the $Y$-axis. at $T^{\prime}$. The normal to the hyperbola at $P(t)$ intersects the $X$-axis at $N$ and the $Y$-axis at $N^{\prime}$. The areas of the triangles $P N T$ and $P N^{\prime} T^{\prime}$ are $\Delta$ and $\Delta^{\prime}$ respectively, then $\frac{1}{\Delta}+\frac{1}{\Delta^{\prime}}$ is

Equation of the chord of the hyperbola $25 x^2-16 y^2=400$ which is bisected at the point $(6,2)$ is

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

The angle between the two asymptotes of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ is

The product of perpendicular distances from any point on the hyperbola $9{x^2} - 16{y^2} = 144$ to its asymptotes is

$P(10,16)$ is a point on the hyperbola having vertices at $(\pm 6,0)$. The equation of the normal to the hyperbola at $P$ is

Let $A$ and $B$ be two fixed points and $P$, another point in the plane, moves in such a way that $k_1 P A+k_2 P B=k_3$, where $k_1, k_2$ and $k_3$ are real constants. The locus of $P$ is
Which one in the options is not true?

The equation $\left| {\sqrt {{x^2} + {{\left( {y - 1} \right)}^2}} - \sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} } \right| = k$ will represent a hyperbola for

The equations of the asymptotes of the hyperbola
$2{x^2} + 5xy + 2{y^2} - 11x - 7y - 4 = 0$ are

The equation of the hyperbola which is referred to its axes as axes of coordinate, and whose distance between the foci is 16 and eccentricity is $\sqrt{2}$, is

The angle between the asymptotes of $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is equal to

A hyperbola has the asymptotes $x + 2y = 3$ and $x - y = 0$ and passes through $\left( {2,1} \right)$. Its centre is

The locus of the point of intersection of the straight lines $\frac{x}{a} + \frac{y}{b} = \lambda$ and $\frac{x}{a} - \frac{y}{b} = \frac{1}{\lambda }$ ( $\lambda$ is a variable) is

Normal at point $(5,3)$ to the rectangular hyperbola $x y-y-2 x-2=0$ meets the curve at the point whose coordinates are

If the equation of the hyperbola is $\frac{y^2}{9}-\frac{x^2}{27}=1$, then

I. the coordinates of the foci are $(0, \pm 6)$

II. the length of the latus rectum is 18 units

III. the eccentricity is $\frac{4}{5}$

A series of hyperbola is drawn having a common transverse axis of length $2 a$. Then the locus of a point $P$ on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymptote, is

The equation of the hyperbola referred to its axes as coordinate axes, whose latus rectum is 4 and eccentricity is 3 is

Equations of the asymptotes of the hyperbola $3 x^2+10 x y+8 y^2+14 x+22 y+7=0$ is

The mid point of the chord $4x - 3y = 5$ of the hyperbola $2{x^2} - 3{y^2} = 12$ is

For different values of $\alpha$, the locus of the point of intersection of the two straight lines $\sqrt 3 x - y - 4\sqrt 3 \alpha = 0\;\,\mathrm{and\;}\sqrt 3 \alpha x + \alpha y - 4\sqrt 3 = 0\;$ is

If the line $y = 2x + \lambda$ be a tangent to the hyperbola $36{x^2} - 25{y^2} = 3600,$ then $\lambda$ is equal to

If the tangents drawn from a point on the hyperbola $x^2-y^2=a^2-b^2$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ make angles $\alpha$ and $\beta$ with the transverse axis of the hyperbola, then