Length of transverse axis of parabola 3{x^2}

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10-2. Parabola, Ellipse, Hyperbola

easy

The length of transverse axis of the parabola $3{x^2} - 4{y^2} = 32$ is

$\frac{{8\sqrt 2 }}{{\sqrt 3 }}$

$\frac{{16\sqrt 2 }}{{\sqrt 3 }}$

$\frac{3}{{32}}$

$\frac{{64}}{3}$

Solution

(a) The given equation may be written as $\frac{{{x^2}}}{{32/2}} – \frac{{{y^2}}}{8} = 1$ or $\frac{{{x^2}}}{{{{\left( {4\sqrt 2 /\sqrt 3 } \right)}^2}}} – \frac{{{y^2}}}{{{{(2\sqrt 2 )}^2}}} = 1$.

Comparing the given equation with $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$,

we get ${a^2} = {\left( {\frac{{4\sqrt 2 }}{{\sqrt 3 }}} \right)^2}$ or $a = \frac{{4\sqrt 2 }}{{\sqrt 3 }}.$

Therefore length of transverse axis of a hyperbola

$ = 2a = 2 \times \frac{{4\sqrt 2 }}{{\sqrt 3 }} = \frac{{8\sqrt 2 }}{{\sqrt 3 }}.$

Standard 11

Mathematics

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $AB$ is

hard

(JEE MAIN-2018)

If the product of the perpendicular distances from any point on the hyperbola $\frac{{{x^2}}}{{{a^2}}}\,\, – \,\,\frac{{{y^2}}}{{{b^2}}}\,\,\, = \,1$ of eccentricity $e =\sqrt 3 \,$ from its asymptotes is equal to $6$, then the length of the transverse axis of the hyperbola is

normal

The difference of the focal distance of any point on the hyperbola $9{x^2} – 16{y^2} = 144$, is

easy

Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, parallel to the straight line $2 x-y=1$. The points of contacts of the tangents on the hyperbola are

$(A)$ $\left(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ $(B)$ $\left(-\frac{9}{2 \sqrt{2}},-\frac{1}{\sqrt{2}}\right)$

$(C)$ $(3 \sqrt{3},-2 \sqrt{2})$ $(D)$ $(-3 \sqrt{3}, 2 \sqrt{2})$

normal

(IIT-2012)

A hyperbola whose transverse axis is along the major axis of then conic, $\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = 4$ and has vertices at the foci of this conic . If the eccentricity of the hyperbola is $\frac{3}{2}$ , then which of the following points does $NOT$ lie on it ?