Hyperbola

If the line y = mx + c is a common tangent to the hyperbola ${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$ and the circle x² + y² = 36, then which one of the following is true?

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

If the tangents drawn to the hyperbola 4y² = x² + 1 intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of AB is :

The locus of the point of intersection of the lines, $\sqrt 2 x - y + 4\sqrt 2 k = 0$ and $\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0$ (k is any non-zero real parameter), is :

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :

The locus of the midpoints of the chord of the circle, x² + y² = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :

The locus of the centroid of the triangle formed by any point P on the hyperbola $16{x^2} - 9{y^2} + 32x + 36y - 164 = 0$, and its foci is :

The locus of the mid points of the chords of the hyperbola x² $-$ y² = 4, which touch the parabola y² = 8x, is :

Let T and C respectively be the transverse and conjugate axes of the hyperbola $16{x^2} - {y^2} + 64x + 4y + 44 = 0$. Then the area of the region above the parabola ${x^2} = y + 4$, below the transverse axis T and on the right of the conjugate axis C is :

Let R be a rectangle given by the lines $x=0, x=2, y=0$ and $y=5$. Let A$(\alpha,0)$ and B$(0,\beta),\alpha\in[0,2]$ and $\beta\in[0,5]$, be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. Then, the mid-point of AB lies on a :

The normal to a curve at $P(x,y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a :

A normal to the hyperbola, 4x² $-$ 9y² = 36 meets the co-ordinate axes $x$ and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the ocus of P is :

If the line y = mx + 7$\sqrt 3$ is normal to the hyperbola ${{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1$ , then a value of m is :

If a hyperbola passes through the point P(10, 16) and it has vertices at (± 6, 0), then the equation of the normal to it at P is :

Let P(3, 3) be a point on the hyperbola, ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. If the normal to it at P intersects the x-axis at (9, 0) and e is its eccentricity, then the ordered pair (a², e²) is equal to :

The point $P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$ lies on the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$ having eccentricity ${{\sqrt 5 } \over 2}$. If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to :

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

Let the tangent drawn to the parabola $y^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x+2 y=5$. Then the normal to the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$ at the point $(\alpha+4, \beta+4)$ does NOT pass through the point :

Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a} > \mathrm{b}$, which passes through the foci of the hyperbola. If $\mathrm{e}_1 \mathrm{e}_2=1$, then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is :