MEDIUM

JEE Main/Advanced

IMPORTANT

Earn 100

$P$ and $Q$ are two variable points on the rectangular hyperbola $x y = c^{2}$ such that the tangents at $Q$ passes through the foot of the ordinate of $P$ . Show that the locus of the point of intersection of tangents at $P$ and $Q$ is a hyperbola with the same asymptotes as the given hyperbola.

Important Questions on Hyperbola

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 12

P, Q, R, S

are four points on a rectangular hyperbola such that the chord

P Q

is perpendicular to the chord

R S

. Prove that each of the four points is the orthocentre of the triangle formed by the other three.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 13

Find the range of parameter '

a

' for which a unique circle will pass through the point of intersection of the rectangular hyperbola

x^{2} - y^{2} = a^{2}

and the parabola

y = x^{2}

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 14

If the normals at

(x_{i}, y_{i}), i = 1, 2, 3, 4

to the rectangular hyperbola

x y = 2

meet at the point

(3, 4)

, then

\frac{\sqrt{2} \sum t_{1} t_{2} t_{3}}{t_{1} t_{2} t_{3} t_{4}}

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 15

The normals at three points

P, Q, R

on a rectangular hyperbola intersect at a point T on the curve. Prove that the centre of the hyperbola is the centroid of

∆ P Q R .

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 16

A normal to the hyperbola

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

meets the axes at

M and N

and lines

MP

and

NP

are drawn perpendicular to axes meeting at

P

. Prove that the locus of

P

is the hyperbola

a^{2} x^{2} - b^{2} y^{2} = {(a^{2} + b^{2})}^{2}

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 18

Show that, if a rectangular hyperbola cut a circle in four points, the centre of mean position of the four points is midway between the centres of the two curves.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 19

Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 21

A straight line is drawn parallel to the conjugate axis of a hyperbola to meet it and the conjugate hyperbola in the points

P

and

Q

respectively. Show that the tangents at

P

and

Q

meet on the curve

\frac{y^{4}}{b^{4}} (\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}}) = \frac{4 x^{2}}{a^{2}}

and that the normals meet on the axis of

x

P and Q are two variable points on the rectangular hyperbola xy=c2 such that the tangents at Q passes through the foot of the ordinate of P. Show that the locus of the point of intersection of tangents at P and Q is a hyperbola with the same asymptotes as the given hyperbola.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

$P$ and $Q$ are two variable points on the rectangular hyperbola $x y = c^{2}$ such that the tangents at $Q$ passes through the foot of the ordinate of $P$ . Show that the locus of the point of intersection of tangents at $P$ and $Q$ is a hyperbola with the same asymptotes as the given hyperbola.