UFM Pure

grandes-ecoles 2013 QV.C.2 Locus and Trajectory Derivation View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$ make a perspective drawing illustrating what precedes.

isi-entrance 2010 Q19 Locus and Trajectory Derivation View

Consider the branch of the rectangular hyperbola $xy = 1$ in the first quadrant. Let $P$ be a fixed point on this curve. The locus of the mid-point of the line segment joining $P$ and an arbitrary point $Q$ on the curve is part of
(a) A hyperbola
(b) A parabola
(c) An ellipse
(d) None of the above.

isi-entrance 2011 Q3 Chord Properties and Midpoint Problems View

Let a line with slope of $60 ^ { \circ }$ be drawn through the focus $F$ of the parabola $y ^ { 2 } = 8 ( x + 2 )$. If the two points of intersection of the line with the parabola are $A$ and $B$ and the perpendicular bisector of the chord $A B$ intersects the $x$-axis at the point $P$, then the length of the segment PF is
(a) $16 / 3$
(b) $8 / 3$
(c) $16 \sqrt{3} / 3$
(d) $8 \sqrt{3}$

isi-entrance 2012 Q10 Locus and Trajectory Derivation View

From a point $P(h,k)$, two tangents are drawn to the parabola $y^2 = 4ax$ which are perpendicular to each other. Find the locus of $P$.

isi-entrance 2013 Q41 4 marks Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x^2 + 4y^2 = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $PC : PD$ equals
(A) 2
(B) $1/2$
(C) 4
(D) $1/4$

isi-entrance 2016 Q41 4 marks Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x^2 + 4y^2 = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $PC : PD$ equals
(A) 2
(B) $1/2$
(C) 4
(D) $1/4$

isi-entrance 2016 Q41 4 marks Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(A) 2
(B) $1 / 2$
(C) 4
(D) $1 / 4$

isi-entrance 2021 Q5 Optimization on Conics View

Let $a , b , c , d > 0$, be any real numbers. Then the maximum possible value of $c x + d y$, over all points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, must be
(A) $\sqrt { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } }$.
(B) $\sqrt { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } }$.
(C) $\sqrt { \frac { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } } { a ^ { 2 } + b ^ { 2 } } }$.
(D) $\sqrt { \frac { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } } { c ^ { 2 } + d ^ { 2 } } }$.

isi-entrance 2022 Q16 Tangent and Normal Line Problems View

Let $y = x + c _ { 1 } , y = x + c _ { 2 }$ be the two tangents to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 1$. What is the value of $\left| c _ { 1 } - c _ { 2 } \right|$?
(A) $\sqrt { 2 }$
(B) $\sqrt { 5 }$
(C) $\frac { \sqrt { 5 } } { 2 }$
(D) 1

isi-entrance 2026 Q9 Tangent and Normal Line Problems View

Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals
(a) 2 .
(B) $1 / 2$.
(C) 4 .
(D) $1 / 4$.

jee-advanced 2007 Q57 Confocal or Related Conic Construction View

A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2 + 4y^2 = 12$. Then its equation is
(A) $x^2\csc^2\theta - y^2\sec^2\theta = 1$
(B) $x^2\sec^2\theta - y^2\csc^2\theta = 1$
(C) $x^2\sin^2\theta - y^2\cos^2\theta = 1$
(D) $x^2\cos^2\theta - y^2\sin^2\theta = 1$

jee-advanced 2008 Q3 Triangle or Quadrilateral Area and Perimeter with Foci View

Consider a branch of the hyperbola
$$x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0$$
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 the area of the triangle $A B C$ is
(A) $1 - \sqrt { \frac { 2 } { 3 } }$
(B) $\sqrt { \frac { 3 } { 2 } } - 1$
(C) $1 + \sqrt { \frac { 2 } { 3 } }$
(D) $\sqrt { \frac { 3 } { 2 } } + 1$

jee-advanced 2008 Q8 Confocal or Related Conic Construction View

Let $P \left( x _ { 1 } , y _ { 1 } \right)$ and $Q \left( x _ { 2 } , y _ { 2 } \right) , y _ { 1 } < 0 , y _ { 2 } < 0$, be the end points of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$. The equations of parabolas with latus rectum $P Q$ are
(A) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(B) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 + \sqrt { 3 }$
(C) $x ^ { 2 } + 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$
(D) $x ^ { 2 } - 2 \sqrt { 3 } \quad y = 3 - \sqrt { 3 }$

jee-advanced 2009 Q22 Locus and Trajectory Derivation View

The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points
(A) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$
(B) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$
(C) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$
(D) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$

jee-advanced 2009 Q25 Triangle or Quadrilateral Area and Perimeter with Foci View

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A , M$ and the origin $O$ is
(A) $\frac { 31 } { 10 }$
(B) $\frac { 29 } { 10 }$
(C) $\frac { 21 } { 10 }$
(D) $\frac { 27 } { 10 }$

jee-advanced 2009 Q25 Equation Determination from Geometric Conditions View

An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then
(A) Equation of ellipse is $x^{2}+2y^{2}=2$
(B) The foci of ellipse are $(\pm1,0)$
(C) Equation of ellipse is $x^{2}+2y^{2}=4$
(D) The foci of ellipse are $(\pm\sqrt{2},0)$

jee-advanced 2009 Q27 Locus and Trajectory Derivation View

The tangent $PT$ and the normal $PN$ to the parabola $y^{2}=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
(A) vertex is $\left(\frac{2a}{3},0\right)$
(B) directrix is $x=0$
(C) latus rectum is $\frac{2a}{3}$
(D) focus is $(a,0)$

jee-advanced 2009 Q40 Conic Identification and Conceptual Properties View

Match the conics in Column I with the statements/expressions in Column II.
Column I
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Column II
(p) The locus of the point $( h , k )$ for which the line $h x + k y = 1$ touches the circle $x ^ { 2 } + y ^ { 2 } = 4$
(q) Points $z$ in the complex plane satisfying $| z + 2 | - | z - 2 | = \pm 3$
(r) Points of the conic have parametric representation $x = \sqrt { 3 } \left( \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } \right) , y = \frac { 2 t } { 1 + t ^ { 2 } }$
(s) The eccentricity of the conic lies in the interval $1 \leq x < \infty$
(t) Points $z$ in the complex plane satisfying $\operatorname { Re } ( z + 1 ) ^ { 2 } = | z | ^ { 2 } + 1$

jee-advanced 2010 Q34 Tangent and Normal Line Problems View

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The coordinates of $A$ and $B$ are
A) $( 3,0 )$ and $( 0,2 )$
B) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$
C) $\left( - \frac { 8 } { 5 } , \frac { 2 \sqrt { 161 } } { 15 } \right)$ and $( 0,2 )$
D) $(3, 0)$ and $\left( - \frac { 9 } { 5 } , \frac { 8 } { 5 } \right)$

jee-advanced 2010 Q37 Circle-Conic Interaction with Tangency or Intersection View

Let A and B be two distinct points on the parabola $\mathrm { y } ^ { 2 } = 4 \mathrm { x }$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining A and B can be
A) $- \frac { 1 } { r }$
B) $\frac { 1 } { r }$
C) $\frac { 2 } { r }$
D) $- \frac { 2 } { \mathrm { r } }$

jee-advanced 2010 Q50 Eccentricity or Asymptote Computation View

The line $2 \mathrm { x } + \mathrm { y } = 1$ is tangent to the hyperbola $\frac { \mathrm { x } ^ { 2 } } { \mathrm { a } ^ { 2 } } - \frac { \mathrm { y } ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

jee-advanced 2011 Q41 Eccentricity or Asymptote Computation View

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 is
(A) $\sqrt { \frac { 5 } { 2 } }$
(B) $\sqrt { \frac { 3 } { 2 } }$
(C) $\sqrt { 2 }$
(D) $\sqrt { 3 }$

jee-advanced 2013 Q55 Focal Chord and Parabola Segment Relations View

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
Length of chord $P Q$ is
(A) $7 a$
(B) $5 a$
(C) $2 a$
(D) $3 a$

jee-advanced 2013 Q56 Focal Chord and Parabola Segment Relations View

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
If chord $P Q$ subtends an angle $\theta$ at the vertex of $y ^ { 2 } = 4 a x$, then $\tan \theta =$
(A) $\frac { 2 } { 3 } \sqrt { 7 }$
(B) $\frac { - 2 } { 3 } \sqrt { 7 }$
(C) $\frac { 2 } { 3 } \sqrt { 5 }$
(D) $\frac { - 2 } { 3 } \sqrt { 5 }$

jee-advanced 2013 Q59 Triangle or Quadrilateral Area and Perimeter with Foci View

A vertical line passing through the point $( h , 0 )$ intersects the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta ( h ) =$ area of the triangle $P Q R$, $\Delta _ { 1 } = \max _ { 1/2 \leq h \leq 1 } \Delta ( h )$ and $\Delta _ { 2 } = \min _ { 1/2 \leq h \leq 1 } \Delta ( h )$, then $\frac { 8 } { \sqrt { 5 } } \Delta _ { 1 } - 8 \Delta _ { 2 } =$

UFM Pure

Sequences and series, recurrence and convergence 375

Roots of polynomials 146

Polar coordinates 22

Conic sections 291

Taylor series 183

Hyperbolic functions 10

Integration using inverse trig and hyperbolic functions 6

Integration with Partial Fractions 7

Vectors: Lines & Planes 293

Vectors: Cross Product & Distances 20

First order differential equations (integrating factor) 60

Complex numbers 2 113

Second order differential equations 127

Systems of differential equations 41