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Hyperbola

56 questions · Mathematics · JEE Main

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Mathematics
Hyperbola

Hyperbola

56 questions · Mathematics · JEE Main

Let one focus of the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be at $(10, 0)$ and the corresponding directrix be $x = \frac{9}{10}$ . If $e$ and $l$ respectively are the eccentricity and the…2025 · 2 Apr · Shift 1 · Q35 · MCQ
Let the product of the focal distances of the point $P (4, 23)$ on the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be 32 . Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$ …2025 · 3 Apr · Shift 1 · Q48 · Numerical
If the equation of the hyperbola with foci $(4, 2)$ and $(8, 2)$ is $3 x^{2} - y^{2} - α x + β y + γ = 0$ , then $α + β + γ$ is equal to .2025 · 3 Apr · Shift 2 · Q47 · Numerical
Let the sum of the focal distances of the point $P (4, 3)$ on the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be $8 \frac{5}{3}$ . If for H , the length of the latus rectum is $l$ and the…2025 · 4 Apr · Shift 2 · Q29 · MCQ
Consider the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ having one of its focus at $P (- 3, 0)$ . If the latus ractum through its other focus subtends a right angle at P and $a^{2} b^{2} = α 2 - β, α, β \in N$ …2025 · 7 Apr · Shift 1 · Q50 · Numerical
Let e1 and e2 be the eccentricities of the ellipse $\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{25} = 1$ and the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1$ , respectively. If b < 5 and e1e2 = 1, then the eccentricity of the ellipse having its…2025 · 7 Apr · Shift 2 · Q34 · MCQ
Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2 a$ and $2 b$ , respectively, and one focus and the corresponding directrix of this hyperbola be $(- 5, 0)$ and $5 x + 9 = 0$ , respectively. If the product…2025 · 7 Apr · Shift 2 · Q49 · Numerical
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 :2025 · 22 Jan · Shift 1 · Q26 · MCQ
Let $H_{1} : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ and $H_{2} : - \frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1$ be two hyperbolas having length of latus rectums $152$ and $125$ …2025 · 24 Jan · Shift 2 · Q50 · Numerical
For $0 < θ < π /2$ , if the eccentricity of the hyperbola $x^{2} - y^{2} cosec^{2} θ = 5$ is $7$ times eccentricity of the ellipse $x^{2} cosec^{2} θ + y^{2} = 5$ , then the value of $θ$ is :2024 · 1 Feb · Shift 1 · Q43 · MCQ
Consider a hyperbola $H$ having centre at the origin and foci on the $x$ -axis. Let $C_{1}$ be the circle touching the hyperbola $H$ and having the centre at the origin. Let $C_{2}$ be the circle…2024 · 4 Apr · Shift 2 · Q31 · MCQ
The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and $x = \pm \frac{4}{3}$ , respectively. Let the line $y - 3 x + 3 = 0$ touch this hyperbola at $(x_{0}, y_{0})$ . If $m$ …2024 · 6 Apr · Shift 2 · Q55 · Numerical
Let $H : \frac{- x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ be the hyperbola, whose eccentricity is $3$ and the length of the latus rectum is $43$ . Suppose the point $(α, 6), α > 0$ lies on $H$ . If $β$ is the product of the…2024 · 8 Apr · Shift 1 · Q37 · MCQ
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 $A (6, 5)$ and passing through the point $S$ . If $O$ …2024 · 8 Apr · Shift 2 · Q54 · Numerical
Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E : \frac{( x - 1 ) ^{2}}{100} + \frac{( y - 1 ) ^{2}}{75} = 1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$ . If the length of the…2024 · 9 Apr · Shift 2 · Q31 · MCQ
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 $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ , which passes through the foci of the hyperbola.…2024 · 27 Jan · Shift 2 · Q38 · MCQ
Let the latus rectum of the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{b ^{2}} = 1$ subtend an angle of $\frac{π}{3}$ at the centre of the hyperbola. If $b^{2}$ is equal to $\frac{l}{m} (1 + n)$ , where $l$ and $m$ …2024 · 30 Jan · Shift 1 · Q56 · Numerical
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 $213$ . Then, the square of the distance of $P$ from the origin is2024 · 30 Jan · Shift 2 · Q43 · MCQ
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 $(2, \frac{14}{3} \frac{2}{5})$ …2024 · 31 Jan · Shift 1 · Q50 · MCQ
Let the foci and length of the latus rectum of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > bb e (\pm 5, 0)$ and $50$ , respectively. Then, the square of the eccentricity of the hyperbola $\frac{x ^{2}}{b ^{2}} - \frac{y ^{2}}{a ^{2} b ^{2}} = 1$ …2024 · 31 Jan · Shift 1 · Q57 · Numerical
Let the eccentricity of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is reciprocal to that of the hyperbola $2 x^{2} - 2 y^{2} = 1$ . If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum…2023 · 6 Apr · Shift 2 · Q38 · Numerical
Let R be a rectangle given by the lines $x = 0, x = 2, y = 0$ and $y = 5$ . Let A $(α, 0)$ and B $(0, β), α \in [0, 2]$ and $β \in [0, 5]$ , be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. Then, the…2023 · 11 Apr · Shift 1 · Q25 · MCQ
Let $H_{n} : \frac{x ^{2}}{1 + n} - \frac{y ^{2}}{3 + n} = 1, n \in N$ . 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…2023 · 11 Apr · Shift 1 · Q42 · Numerical
Let T and C respectively be the transverse and conjugate axes of the hyperbola $16 x^{2} - y^{2} + 64 x + 4 y + 44 = 0$ . Then the area of the region above the parabola $x^{2} = y + 4$ , below the transverse axis T and on the right of the…2023 · 25 Jan · Shift 2 · Q29 · MCQ
Let $H$ be the hyperbola, whose foci are $(1 \pm 2, 0)$ and eccentricity is $2$ . Then the length of its latus rectum is :2023 · 31 Jan · Shift 2 · Q34 · MCQ
Let the hyperbola $H : \frac{x ^{2}}{a ^{2}} - y^{2} = 1$ 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…2022 · 24 Jun · Shift 2 · Q44 · Numerical
Let the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{7} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{α} = \frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is :2022 · 25 Jul · Shift 2 · Q32 · MCQ
If the line $x - 1 = 0$ is a directrix of the hyperbola $k x^{2} - y^{2} = 6$ , then the hyperbola passes through the point :2022 · 26 Jul · Shift 2 · Q30 · MCQ
An ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ passes through the vertices of the hyperbola $H : \frac{x ^{2}}{49} - \frac{y ^{2}}{64} = - 1$ . Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate…2022 · 27 Jul · Shift 1 · Q41 · Numerical
Let the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ pass through the point $(22, - 22)$ . A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the…2022 · 28 Jul · Shift 2 · Q32 · MCQ
Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . Let e' and l' respectively be the eccentricity and length of…2022 · 28 Jun · Shift 2 · Q34 · MCQ
Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ , a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4 (22 + 14)$ . If the eccentricity H is $\frac{11}{2}$ …2022 · 29 Jun · Shift 1 · Q40 · Numerical
The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, $\frac{x ^{2}}{9} - \frac{y ^{2}}{16} = 1$ is :2021 · 16 Mar · Shift 1 · Q30 · MCQ
The locus of the point of intersection of the lines $(3) k x + k y - 43 = 0$ and $3 x - y - 4 (3) k = 0$ is a conic, whose eccentricity is .2021 · 25 Feb · Shift 1 · Q37 · Numerical
A hyperbola passes through the foci of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their…2021 · 25 Feb · Shift 2 · Q40 · MCQ
The locus of the centroid of the triangle formed by any point P on the hyperbola $16 x^{2} - 9 y^{2} + 32 x + 36 y - 164 = 0$ , and its foci is :2021 · 25 Jul · Shift 1 · Q26 · MCQ
The locus of the mid points of the chords of the hyperbola x2 $-$ y2 = 4, which touch the parabola y2 = 8x, is :2021 · 26 Aug · Shift 2 · Q32 · MCQ
For some $θ \in (0, \frac{π}{2})$ , if the eccentricity of the hyperbola, x2–y2sec2 $θ$ = 10 is $5$ times the eccentricity of the ellipse, x2sec2 $θ$ + y2 = 5, then the length of the latus rectum of the…2020 · 2 Sep · Shift 2 · Q39 · MCQ
A hyperbola having the transverse axis of length $2$ has the same foci as that of the ellipse 3x2 + 4y2 = 12, then this hyperbola does not pass through which of the following points?2020 · 3 Sep · Shift 1 · Q33 · MCQ
Let e1 and e2 be the eccentricities of the ellipse, $\frac{x ^{2}}{25} + \frac{y ^{2}}{b ^{2}} = 1$ (b < 5) and the hyperbola, $\frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1$ respectively satisfying e1e2 = 1. If $α$ and $β$ …2020 · 3 Sep · Shift 2 · Q27 · MCQ
If e1 and e2 are the eccentricities of the ellipse, $\frac{x ^{2}}{18} + \frac{y ^{2}}{4} = 1$ and the hyperbola, $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ respectively and (e1, e2) is a point on the ellipse, 15x2 + 3y2 = k, then k…2020 · 9 Jan · Shift 1 · Q40 · MCQ
Let $0 < θ < \frac{π}{2}$ . If the eccentricity of the hyperbola $\frac{x ^{2}}{c o s ^{2} θ} - \frac{y ^{2}}{s i n ^{2} θ}$ = 1 is greater than 2, then the length of its latus rectum lies in the interval :2019 · 9 Jan · Shift 1 · Q29 · MCQ
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 :2019 · 9 Jan · Shift 2 · Q28 · MCQ
If a directrix of a hyperbola centred at the origin and passing through the point (4, –2 $3$ ) is 5x = 4 $5$ and its eccentricity is e, then :2019 · 10 Apr · Shift 1 · Q41 · MCQ
If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is :2019 · 10 Apr · Shift 2 · Q41 · MCQ
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 :2019 · 11 Jan · Shift 2 · Q22 · MCQ
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 :2019 · 11 Jan · Shift 2 · Q38 · MCQ
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?2019 · 12 Jan · Shift 1 · Q30 · MCQ
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 :2018 · 15 Apr · Shift 1 · Q34 · MCQ
The locus of the point of intersection of the lines, $2 x - y + 42 k = 0$ and $2 k x + k y - 42 = 0$ (k is any non-zero real parameter), is :2018 · 16 Apr · Shift 1 · Q47 · MCQ
The locus of the point of intersection of the straight lines, tx $-$ 2y $-$ 3t = 0 x $-$ 2ty + 3 = 0 (t $\in$ R), is :2017 · 8 Apr · Shift 1 · Q44 · MCQ
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 −…2016 · 9 Apr · Shift 1 · Q33 · MCQ
A hyperbola whose transverse axis is along the major axis of the 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…2016 · 10 Apr · Shift 1 · Q34 · MCQ
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 :2016 · Shift 0 · Q31 · MCQ
For the Hyperbola $\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$ , which of the following remains constant when $α$ varies $=$ ?2007 · Shift 0 · Q55 · MCQ
The foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25}$ coincide. Then the value of $b^{2}$ is :2003 · Shift 0 · Q114 · MCQ