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Hyperbola

16 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Hyperbola

Hyperbola

16 questions · Mathematics · JEE Advanced

Consider the hyperbola $\frac{x ^{2}}{100} - \frac{y ^{2}}{64} = 1$ with foci at $S$ and $S_{1}$ , where $S$ lies on the positive $x$ -axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $∠ S P S_{1} = α$ , with $α < \frac{π}{2}$ …2022 · Shift 2 · Q25 · Numerical
Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . Suppose the tangent to the hyperbola at P…2020 · Shift 2 · Q26 · Multiple correct
Let T be the line passing through the points P( $-$ 2, 7) and Q(2, $-$ 5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point,…2018 · Shift 2 · Q20 · Multiple correct
Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ , where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60 $^{\circ}$ at one of its vertices N. Let the area of the $Δ$ LMN be $43$ … Includes table2018 · Shift 2 · Q35 · MCQ
If $2 x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{16} = 1$ then which of the following CANNOT be sides of a right angled triangle?2017 · Shift 1 · Q22 · Multiple correct
By appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. For $a = 2$ , if a tangent is… Includes table2017 · Shift 1 · Q31 · MCQ
By appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. The tangent to a suitable conic… Includes table2017 · Shift 1 · Q32 · MCQ
Consider the hyperbola $H : x^{2} - y^{2} = 1$ and a circle $S$ with center $N (x_{2}, 0)$ . Suppose that $H$ and $S$ touch each other at a point $P (x_{1}, y_{1})$ with $x_{1} > 1$ and $y_{1} > 0$ . The…2015 · Shift 2 · Q30 · Multiple correct
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 contact of the tangents on the hyperbola are2012 · Shift 1 · Q28 · Multiple correct
Let the eccentricity of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be reciprocal to that of the ellipse $x^{2} + 4 y^{2} = 4$ . If the hyperbola passes through a focus of the ellipse, then2011 · Shift 1 · Q31 · Multiple correct
Let $P (6, 3)$ be a point on the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . If the normal at the point $P$ intersects the $x$ -axis at $(9, 0)$ , then the eccentricity of the hyperbola is2011 · Shift 2 · Q27 · MCQ
The circle $x^{2} + y^{2} - 8 x = 0$ and hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ intersect at the points $A$ and $B$ . Equation of a common tangent with positive slope to the circle as well as to the hyperbola is2010 · Shift 1 · Q36 · MCQ
The circle $x^{2} + y^{2} - 8 x = 0$ and hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ intersect at the points $A$ and $B$ . Equation of the circle with $A B$ as its diameter is2010 · Shift 1 · Q37 · MCQ
The line $2 x + y = 1$ is tangent to the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . If this line passes through the point of intersection of the nearest directrix and the $x$ -axis, then the eccentricity of the…2010 · Shift 1 · Q38 · Numerical
Consider a branch of the hyperbola $x^{2} - 2 y^{2} - 22 x - 42 y - 6 = 0$ with vertex at the point $A$ . Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$ , then…2008 · Shift 2 · Q36 · MCQ
A hyperbola, having the transverse axis of the length $2 sin θ$ , is confocal with the ellipse $3 x^{2} + 4 y^{2} = 12$ . Then its equation is2007 · Shift 1 · Q28 · MCQ