JEE Hyperbola PYQs with Video Solutions, Download PDF

The area (in square units) of the triangle formed by the asymptotes of the hyperbola $$xy - 3x - 2y = 0$$ and the line $$x + y = 2$$ is:

If the foci of a hyperbola are same as that of the ellipse $$\frac{x^2}{9} + \frac{y^2}{25} = 1$$ and the eccentricity of the hyperbola is $$\frac{15}{8}$$ times the eccentricity of the ellipse, then the smaller focal distance of the point $$\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$$ on the hyperbola, is equal to

The length of the latus rectum and directrices of a hyperbola with eccentricity $$e$$ are 9 and $$x = \pm \frac{4}{\sqrt{13}}$$, respectively. Let the line $$y - \sqrt{3}x + \sqrt{3} = 0$$ touch this hyperbola at $$(x_0, y_0)$$. If $$m$$ is the product of the focal distances of the point $$(x_0, y_0)$$, then $$4e^2 + m$$ is equal to ___________

Let T and C respectively, be the transverse and conjugate axes of the hyperbola $$16x^2 - y^2 + 64x + 4y + 44 = 0$$. 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 the foci of a hyperbola coincide with the foci of the ellipse $$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$$. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

Let $$P(10, 2\sqrt{15})$$ be a point on the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$, whose foci are S and S'. if the length of its latus rectum is 8, then the square of the area of $$\Delta PSS'$$ is equal to:

Let $$H_n: \frac{x^2}{1+n} - \frac{y^2}{3+n} = 1$$, $$n \in \mathbb{N}$$. Let $$k$$ be the smallest even value of $$n$$ such that the eccentricity of $$H_k$$ is a rational number. If $$l$$ is the length of the latus rectum of $$H_k$$, then $$21l$$ is equal to _______

A hyperbola having the transverse axis of length $$\sqrt{2}$$ has the same foci as that of the ellipse, $$3x^2 + 4y^2 = 12$$ then this hyperbola does not pass through which of the following points?

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

If the line $$x - 1 = 0$$ is a directrix of the hyperbola $$kx^2 - y^2 = 6$$, then the hyperbola passes through the point

The locus of a point $$P(\alpha, \beta)$$ moving under the condition that the line $$y = \alpha x + \beta$$ is a tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ is

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:

Let $$H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be the hyperbola, whose eccentricity is $$\sqrt{3}$$ and the length of the latus rectum is $$4\sqrt{3}$$. Suppose the point $$(\alpha, 6), \alpha > 0$$ lies on $$H$$. If $$\beta$$ is the product of the focal distances of the point $$(\alpha, 6)$$, then $$\alpha^2 + \beta$$ is equal to

Let P($$3\sec\theta, 2\tan\theta$$) and Q($$3\sec\phi, 2\tan\phi$$) where $$\theta + \phi = \frac{\pi}{2}$$, be two distinct points on the hyperbola $$\frac{x^2}{9} - \frac{y^2}{4} = 1$$. Then the ordinate of the point of intersection of the normals at P and Q is:

If $$5x + 9 = 0$$ is the directrix of the hyperbola $$16x^2 - 9y^2 = 144$$, then its corresponding focus is:

Consider a hyperbola $$H : x^2 - 2y^2 = 4$$. Let the tangent at a point $$P(4, \sqrt{6})$$ meet the x-axis at $$Q$$ and latus rectum at $$R(x_1, y_1)$$, $$x_1 > 0$$. If $$F$$ is a focus of $$H$$ which is nearer to the point $$P$$, then the area of $$\triangle QFR$$ (in sq. units) is equal to

If the line $$\alpha x + 2y = 1$$, where $$\alpha \in R $$, does not meet the hyperbola $$x^{2}-9y^{2}=9$$, then a possible value of $$\alpha$$ is:

Consider the ellipse $$E$$ given by $$\dfrac{x^2}{18}+\dfrac{y^2}{12}=1$$. Let $$H$$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $$E$$ and whose foci are the same as that of $$E$$. Let $$P$$ and $$Q$$ be the points of intersection of $$H$$ and the parabola $$\sqrt{5}\,y=x^2$$ in the first quadrant. Let $$d$$ be the distance between $$P$$ and $$Q$$.

If $$a$$ and $$b$$ are the integers such that $$d^2=a+b\sqrt{5}$$, then the value of $$a-b$$ is ___.

Let S be the focus of the hyperbola $$\frac{x^2}{3} - \frac{y^2}{5} = 1$$, on the positive x-axis. Let C be the circle with its centre at $$A(\sqrt{6}, \sqrt{5})$$ and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to _____

Let the foci of a hyperbola be $$(1, 4)$$ and $$(1, -12).$$ If it passes through the point $$(1, 6)$$, then the length of its latus-rectum is :

For the hyperbola $$\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1$$, which of the following remains constant when $$\alpha$$ varies?

If the eccentricity of a hyperbola $$\dfrac{x^2}{9} - \dfrac{y^2}{b^2} = 1$$, which passes through $$(K, 2)$$, is $$\dfrac{\sqrt{13}}{3}$$, then the value of $$K^2$$ is

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be $$(-5, 0)$$ and $$5x + 9 = 0$$, respectively. If the product of the focal distances of a point $$\left(\alpha, 2\sqrt{5}\right)$$ on the hyperbola is p, then 4p is equal to _____.

Let one focus of the hyperbola H: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be at $$(\sqrt{10}, 0)$$ and the corresponding directrix be $$x = \frac{9}{\sqrt{10}}$$. If $$e$$ and $$l$$ respectively are the eccentricity and the length of the latus rectum of H, then $$9(e^2 + l)$$ is equal to:

If the equation of the hyperbola with foci $$(4, 2)$$ and $$(8, 2)$$ is $$3x^2 - y^2 - \alpha x + \beta y + \gamma = 0$$, then $$\alpha + \beta + \gamma$$ is equal to ________.

If the normal at the point $$P$$ on the hyperbola $$\frac{x^2}{16}-\frac{y^2}{9}=1$$ meets the coordinate axes at $$A$$ and $$B,$$ and the midpoint of $$AB$$ is $$M$$, then the locus of $$M$$ is:

Let the domain of the function $$f(x)=\log_{3}\log_{5}\log_{7}(9x-x^{2}-13)$$ be the interval (m, n). Let the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ have eccentricity $$\frac{n}{3}$$ and the length of the latus rectum $$\frac{8m}{3}$$. Then $$b^{2}-a^{2}$$ is equal to:

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

Let the foci of the ellipse $$\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1$$ and the hyperbola $$\dfrac{x^2}{144} - \dfrac{y^2}{\alpha} = \dfrac{1}{25}$$ coincide. Then the length of the latus rectum of the hyperbola is:

Let $$P(x_0, y_0)$$ be the point on the hyperbola $$3x^2 - 4y^2 = 36$$, which is nearest to the line $$3x + 2y = 1$$. Then $$\sqrt{2}(y_0 - x_0)$$ is equal to :