2022
JEE Mains
Numerical
JEE Main 2022 (Online) 24th June Evening Shift
Let the hyperbola $H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$ and the ellipse $E:3{x^2} + 4{y^2} = 12$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If ${e_H}$ and ${e_E}$ are the eccentricities of H and E respectively, then the value of $12\left( {e_H^2 + e_E^2} \right)$ is equal to ___________.
Correct Answer: 42
Explanation:
$\because$ $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over 1} = 1$
$\therefore$ Length of latus rectum $ = {2 \over a}$
$E:{{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$
Length of latus rectum $ = {6 \over 2} = 3$
$\because$ ${2 \over a} = 3 \Rightarrow a = {2 \over 3}$
$\therefore$ $12\left( {e_H^2 + e_E^2} \right) = 12\left( {1 + {9 \over 4}} \right) + \left( {1 - {3 \over 4}} \right) = 42$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Evening Shift
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 :
A.
$4\sqrt 3 $
B.
6
C.
$6\sqrt 3 $
D.
$3\sqrt 6 $
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th August Evening Shift
The locus of the mid points of the chords of the hyperbola x2 $-$ y2 = 4, which touch the parabola y2 = 8x, is :
A.
y3(x $-$ 2) = x2
B.
x3(x $-$ 2) = y2
C.
y2(x $-$ 2) = x3
D.
x2(x $-$ 2) = y3
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th July Morning Shift
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 :
A.
$16{x^2} - 9{y^2} + 32x + 36y - 36 = 0$
B.
$9{x^2} - 16{y^2} + 36x + 32y - 144 = 0$
C.
$16{x^2} - 9{y^2} + 32x + 36y - 144 = 0$
D.
$9{x^2} - 16{y^2} + 36x + 32y - 36 = 0$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 22th July Evening Shift
Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $-$ y2 = 3. If L is also a tangent to the parabola y2 = $\alpha$x, then $\alpha$ is equal to :
A.
12
B.
$-$12
C.
24
D.
$-$24
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Consider a hyperbola H : x2 $-$ 2y2 = 4. Let the tangent at a
point P(4, ${\sqrt 6 }$) meet the x-axis at Q and latus rectum at R(x1, y1), x1 > 0. If F is a focus of H which is nearer to the point P, then the area of $\Delta$QFR is equal to :
point P(4, ${\sqrt 6 }$) meet the x-axis at Q and latus rectum at R(x1, y1), x1 > 0. If F is a focus of H which is nearer to the point P, then the area of $\Delta$QFR is equal to :
A.
${\sqrt 6 }$ $-$ 1
B.
${7 \over {\sqrt 6 }}$ $-$ 2
C.
${4\sqrt 6 }$ $-$ 1
D.
${4\sqrt 6 }$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 16th March Morning Shift
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 :
A.
(x2 + y2)2 $-$ 9x2 + 16y2 = 0
B.
(x2 + y2)2 $-$ 9x2 + 144y2 = 0
C.
(x2 + y2)2 $-$ 16x2 + 9y2 = 0
D.
(x2 + y2)2 $-$ 9x2 $-$ 16y2 = 0
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th February Evening Shift
A hyperbola passes through the foci of the ellipse ${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :
A.
${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$
B.
${{{x^2}} \over 9} - {{{y^2}} \over 16} = 1$
C.
${{{x^2}} \over 9} - {{{y^2}} \over 25} = 1$
D.
x2 $-$ y2 = 9
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th August Evening Shift
Let A (sec$\theta$, 2tan$\theta$) and B (sec$\phi$, 2tan$\phi$), where $\theta$ + $\phi$ = $\pi$/2, be two points on the hyperbola 2x2 $-$ y2 = 2. If ($\alpha$, $\beta$) is the point of the intersection of the normals to the hyperbola at A and B, then (2$\beta$)2 is equal to ____________.
Correct Answer: Bonus
Explanation:
Since, point A (sec$\theta$, 2tan$\theta$) lies on the hyperbola 2x2 $-$ y2 = 2
Therefore, 2sec2$\theta$ $-$ 4tan2$\theta$ = 2
$\Rightarrow$ 2 + 2tan2$\theta$ $-$ 4tan2$\theta$ = 2
$\Rightarrow$ tan$\theta$ = 0 $\Rightarrow$ $\theta$ = 0
Similarly, for point B, we will get $\phi$ = 0.
but according to question $\theta$ + $\phi$ = ${\pi \over 2}$ which is not possible.
Hence, it must be a 'BONUS'.
Therefore, 2sec2$\theta$ $-$ 4tan2$\theta$ = 2
$\Rightarrow$ 2 + 2tan2$\theta$ $-$ 4tan2$\theta$ = 2
$\Rightarrow$ tan$\theta$ = 0 $\Rightarrow$ $\theta$ = 0
Similarly, for point B, we will get $\phi$ = 0.
but according to question $\theta$ + $\phi$ = ${\pi \over 2}$ which is not possible.
Hence, it must be a 'BONUS'.
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 25th February Morning Shift
The locus of the point of intersection of the lines $\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$ and $\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0$ is a conic, whose eccentricity is _________.
Correct Answer: 2
Explanation:
$\sqrt 3 kx + ky = 4\sqrt 3 $ ........(1)
$\sqrt 3 kx - ky = 4\sqrt 3 {k^2}$ ....... (2)
Adding equation (1) & (2)
$2\sqrt 3 kx = 4\sqrt 3 ({k^2} + 1)$
$x = 2\left( {k + {1 \over k}} \right)$ ......... (3)
Substracting equation (1) & (2)
$y = 2\sqrt 3 \left( {{1 \over k} - k} \right)$ ........(4)
$\therefore$ ${{{x^2}} \over 4} - {{{y^2}} \over {12}} = 4$
${{{x^2}} \over {16}} - {{{y^2}} \over {48}} = 1$ (Hyperbola)
$ \therefore $ ${e^2} = 1 + {{48} \over {16}}$
$ \Rightarrow $ $e = 2$
$\sqrt 3 kx - ky = 4\sqrt 3 {k^2}$ ....... (2)
Adding equation (1) & (2)
$2\sqrt 3 kx = 4\sqrt 3 ({k^2} + 1)$
$x = 2\left( {k + {1 \over k}} \right)$ ......... (3)
Substracting equation (1) & (2)
$y = 2\sqrt 3 \left( {{1 \over k} - k} \right)$ ........(4)
$\therefore$ ${{{x^2}} \over 4} - {{{y^2}} \over {12}} = 4$
${{{x^2}} \over {16}} - {{{y^2}} \over {48}} = 1$ (Hyperbola)
$ \therefore $ ${e^2} = 1 + {{48} \over {16}}$
$ \Rightarrow $ $e = 2$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 5th September Evening Slot
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?
${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$ and the circle x2 + y2 = 36, then which one of the following is true?
A.
5m = 4
B.
8m + 5 = 0
C.
c2 = 369
D.
4c2 = 369
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 4th September Morning Slot
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 :
${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. 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 :
A.
$\left( {{9 \over 2},2} \right)$
B.
$\left( {{3 \over 2},2} \right)$
C.
(9,3)
D.
$\left( {{9 \over 2},3} \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
Let e1
and e2
be the eccentricities of the
ellipse,
${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$(b < 5) and the hyperbola,
${{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1$ respectively satisfying e1e2 = 1. If $\alpha $
and $\beta $ are the distances between the foci of the
ellipse and the foci of the hyperbola
respectively, then the ordered pair ($\alpha $, $\beta $) is equal to :
${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$(b < 5) and the hyperbola,
${{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1$ respectively satisfying e1e2 = 1. If $\alpha $
and $\beta $ are the distances between the foci of the
ellipse and the foci of the hyperbola
respectively, then the ordered pair ($\alpha $, $\beta $) is equal to :
A.
(8, 10)
B.
(8, 12)
C.
$\left( {{{24} \over 5},10} \right)$
D.
$\left( {{{20} \over 3},12} \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
A hyperbola having the transverse axis of
length
$\sqrt 2 $ has the same foci as that of the ellipse
3x2 + 4y2 = 12, then this hyperbola does not
pass through which of the following points?
A.
$\left( {1, - {1 \over {\sqrt 2 }}} \right)$
B.
$\left( {\sqrt {{3 \over 2}} ,{1 \over {\sqrt 2 }}} \right)$
C.
$\left( { - \sqrt {{3 \over 2}} ,1} \right)$
D.
$\left( {{1 \over {\sqrt 2 }},0} \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
For some $\theta \in \left( {0,{\pi \over 2}} \right)$, if the eccentricity of the
hyperbola, x2–y2sec2$\theta $ = 10 is $\sqrt 5 $ times the
eccentricity of the ellipse, x2sec2$\theta $ + y2 = 5, then the length of the latus rectum of the ellipse, is :
hyperbola, x2–y2sec2$\theta $ = 10 is $\sqrt 5 $ times the
eccentricity of the ellipse, x2sec2$\theta $ + y2 = 5, then the length of the latus rectum of the ellipse, is :
A.
$\sqrt {30} $
B.
$2\sqrt 6 $
C.
${{4\sqrt 5 } \over 3}$
D.
${{2\sqrt 5 } \over 3}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
A line parallel to the straight line 2x – y = 0 is
tangent to the hyperbola
${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$ at the point $\left( {{x_1},{y_1}} \right)$. Then $x_1^2 + 5y_1^2$ is equal to :
${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$ at the point $\left( {{x_1},{y_1}} \right)$. Then $x_1^2 + 5y_1^2$ is equal to :
A.
5
B.
6
C.
10
D.
8
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Morning Slot
If e1 and e2 are the eccentricities of the ellipse,
${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$ and the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ respectively and (e1, e2) is a point on the ellipse,
15x2 + 3y2 = k, then k is equal to :
A.
17
B.
16
C.
15
D.
14
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Evening Slot
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 :
A.
2x + 5y = 100
B.
x + 3y = 58
C.
x + 2y = 42
D.
3x + 4y = 94
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Evening Slot
If 5x + 9 = 0 is the directrix of the hyperbola 16x2
– 9y2
= 144, then its corresponding focus is :
A.
$\left( {{5 \over 3},0} \right)$
B.
(5, 0)
C.
(- 5, 0)
D.
$\left( { - {5 \over 3},0} \right)$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Morning Slot
If a directrix of a hyperbola centred at the origin and passing through the point (4, –2$\sqrt 3 $ ) is 5x = 4$\sqrt 5 $ and
its eccentricity is e, then :
A.
4e4 – 24e2 + 27 = 0
B.
4e4 – 24e2 + 35 = 0
C.
4e4 – 12e2 - 27 = 0
D.
4e4 + 8e2 - 35 = 0
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Morning Slot
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 :
A.
${3 \over {\sqrt 5 }}$
B.
${{\sqrt 15 } \over 2}$
C.
${{\sqrt 5 } \over 2}$
D.
${2 \over {\sqrt 5 }}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Evening Slot
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 :
A.
2x – y – 2 = 0
B.
3x – 2y = 0
C.
2x – 3y + 10 = 0
D.
x – 2y + 8 = 0
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th January Morning Slot
If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this hyperbola?
A.
$\left( {6,5\sqrt 2 } \right)$
B.
$\left( {2\sqrt 6 ,5} \right)$
C.
$\left( { - 6,2\sqrt {10} } \right)$
D.
$\left( {4,\sqrt {15} } \right)$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
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 :
A.
an ellipse
B.
a parabola
C.
a hyperbola
D.
a straight line
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :
A.
${{13} \over 6}$
B.
2
C.
${{13} \over 12}$
D.
${{13} \over 8}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is :
A.
x + y + 1 = 0
B.
4x + 2y + 1 = 0
C.
x – 2y + 4 = 0
D.
x + 2y + 4 = 0
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th January Morning Slot
The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is :
A.
x $-$ y + 9 = 0
B.
x $-$ y $-$ 3 = 0
C.
x $-$ y + 1 = 0
D.
x $-$ y + 7 = 0
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Evening Slot
A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity of the hyperbola is :
A.
${3 \over 2}$
B.
$\sqrt 3 $
C.
2
D.
${2 \over {\sqrt 3 }}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Morning Slot
Let $0 < \theta < {\pi \over 2}$. If the eccentricity of the
hyperbola ${{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}\theta }}$ = 1 is greater
than 2, then the length of its latus rectum lies in the interval :
hyperbola ${{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}\theta }}$ = 1 is greater
than 2, then the length of its latus rectum lies in the interval :
A.
(3, $\infty $)
B.
$\left( {{3 \over 2},2} \right]$
C.
$\left( {1,{3 \over 2}} \right]$
D.
$\left( {2,3} \right]$
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 16th April Morning Slot
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.
an ellipse whose eccentricity is ${1 \over {\sqrt 3 }}.$
B.
an ellipse with length of its major axis $8\sqrt 2 .$
C.
a hyperbola whose eccentricity is $\sqrt 3 .$
D.
a hyperbola with length of its transverse axis $8\sqrt 2 .$
2018
JEE Mains
MCQ
JEE Main 2018 (Offline)
Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q.
If these tangents intersect at the point T(0, 3) then the area (in sq. units) of $\Delta $PTQ is :
If these tangents intersect at the point T(0, 3) then the area (in sq. units) of $\Delta $PTQ is :
A.
$36\sqrt 5 $
B.
$45\sqrt 5 $
C.
$54\sqrt 3 $
D.
$60\sqrt 3 $
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Evening Slot
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 :
A.
4x2 + 9y2 = 121
B.
9x2 + 4y2 = 169
C.
4x2 $-$ 9y2 = 121
D.
9x2 $-$ 4y2 = 169
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Morning Slot
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 :
A.
x2 $-$ 4y2 + 16x2y2 = 0
B.
x2 $-$ 4y2 $-$ 16x2y2 = 0
C.
4x2 $-$ y2 + 16x2y2 = 0
D.
4x2 $-$ y2 $-$ 16x2y2 = 0
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
The locus of the point of intersection of the straight lines,
tx $-$ 2y $-$ 3t = 0
x $-$ 2ty + 3 = 0 (t $ \in $ R), is :
tx $-$ 2y $-$ 3t = 0
x $-$ 2ty + 3 = 0 (t $ \in $ R), is :
A.
an ellipse with eccentricity ${2 \over {\sqrt 5 }}$
B.
an ellipse with the length of major axis 6
C.
a hyperbola with eccentricity $\sqrt 5 $
D.
a hyperbola with the length of conjugate axis 3
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
A hyperbola passes through the point P$\left( {\sqrt 2 ,\sqrt 3 } \right)$ and has foci at $\left( { \pm 2,0} \right)$. Then the tangent to this hyperbola at P also passes through the point :
A.
$\left( {2\sqrt 2 ,3\sqrt 3 } \right)$
B.
$\left( {\sqrt 3 ,\sqrt 2 } \right)$
C.
$\left( { - \sqrt 2 , - \sqrt 3 } \right)$
D.
$\left( {3\sqrt 2 ,2\sqrt 3 } \right)$
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 10th April Morning Slot
A hyperbola whose transverse axis is along the major axis of the conic, ${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is ${3 \over 2},$ then which of the following points does NOT lie on it?
A.
(0, 2)
B.
$\left( {\sqrt 5 ,2\sqrt 2 } \right)$
C.
$\left( {\sqrt {10} ,2\sqrt 3 } \right)$
D.
$\left( {5,2\sqrt 3 } \right)$
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 9th April Morning Slot
Let a and b respectively be the semitransverse and semi-conjugate axes of a
hyperbola whose eccentricity satisfies the equation 9e2 − 18e + 5 = 0. If S(5, 0) is a focus and 5x = 9 is the corresponding directrix of this hyperbola, then a2 − b2 is equal to :
A.
7
B.
$-$ 7
C.
5
D.
$-$ 5
2016
JEE Mains
MCQ
JEE Main 2016 (Offline)
The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci, is :
A.
${2 \over {\sqrt 3 }}$
B.
${\sqrt 3 }$
C.
${{4 \over 3}}$
D.
${4 \over {\sqrt 3 }}$
2007
JEE Mains
MCQ
AIEEE 2007
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.
circle
B.
hyperbola
C.
ellipse
D.
parabola
2007
JEE Mains
MCQ
AIEEE 2007
For the Hyperbola ${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$ , which of the following remains constant when $\alpha $ varies$=$?
A.
abscissae of vertices
B.
abscissae of foci
C.
eccentricity
D.
directrix.
2005
JEE Mains
MCQ
AIEEE 2005
The locus of a point $P\left( {\alpha ,\beta } \right)$ moving under the condition that the line $y = \alpha x + \beta $ is tangent to the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$ is :
A.
an ellipse
B.
a circle
C.
a parabola
D.
a hyperbola
2003
JEE Mains
MCQ
AIEEE 2003
The foci of the ellipse ${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$ and the hyperbola ${{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}}$ coincide. Then the value of ${b^2}$ is :
A.
$9$
B.
$1$
C.
$5$
D.
$7$