Hyperbola Questions (161)

A normal to the hyperbola \(\frac{x^2}{4} - \frac{y^2}{1} = 1\) has equal intercepts on positive x and positive y-axes. If this normal touches the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), then \(3(a^2 + b^2)\) is equal to:

Let \(a\) and \(b\) be any two numbers satisfying \(\dfrac{1}{a^2} + \dfrac{1}{b^2} = \dfrac{1}{4}\). Then, the foot of perpendicular from the origin on the variable line, \(\dfrac{x}{a} + \dfrac{y}{b} = 1\), lies on

Let \(P\) be a point on the hyperbola \(H: \frac{x^{2}}{9}-\frac{y^{2}}{4}=1\), in the first quadrant such that the area of triangle formed by \(P\) and the two foci of \(H\) is \(2 \sqrt{13}\). Then, the square of the distance of \(P\) from the origin is

For some $a, b, c \in \mathbb{N}$, let $f(x) = ax - 3$ and $g(x) = x^b + c$, $x \in \mathbb{R}$. If $(f \circ g)^{-1}(x) = \left(\dfrac{x-7}{2}\right)^{1/3}$, then $(f \circ g)(ac) + (g \circ f)(b)$ is equal to ______.

Let H be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is ______.

The equation of the chord joining two points \((x_1, y_1)\) and \((x_2, y_2)\) on the rectangular hyperbola \(xy = c^2\) is

$(C) 0.75$

Let the foci of a hyperbola be \((1,14)\) and \((1,-12)\). If it passes through the point \((1,6)\), then the length of its latus rectum is:

If two points \(P\) and \(Q\) on the hyperbola \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\), whose centre \(C\) be such that \(CP\) is perpendicular to \(CQ\), \(a

The tangent to the hyperbola y = (x + 9)/(x − 5) passing through the origin is

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 \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), \(a \gt b\), which passes through the foci of the hyperbola. If \(e_{1} e_{2}=1\), then the length of the chord of the ellipse parallel to the \(x\)-axis and passing through \((0,2)\) is: