P is a variable point on the hyperbola in the form x 2 a 2 - y 2 b 2 = 1 , whose vertex A is a , 0 . Show that the locus of the mid point of A P is 2 x - a 2 a 2 - 4 y 2 b 2 = 1 :

Given: P is a variable point on the hyperbola, the equation of the hyperbola is x 2 a 2 - y 2 b 2 = 1 , vertex of the hyperbola is A a , 0 . Let us assume the variable point P on the hyperbola is a sec θ , b tan θ . And, A a , 0 . Let, the mid-point is x , y . So, x = a sec θ + a 2 and y = b tan θ + 0 2 ⇒ 2 x - a = a sec θ and 2 y = b tan θ ⇒ 2 x - a a = sec θ . . . . . 1 and 2 y b = tan θ . . . . . . 2 We also know the property that sec 2 θ - tan 2 θ = 1 . By putting the values of 1 and 2 , 2 x - a a 2 - 2 y b 2 = 1 ⇒ 2 x - a 2 a 2 - 4 y 2 b 2 = 1 Hence, proved.

HARD

JEE Main/Advanced

IMPORTANT

Earn 100

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 $y = x^{2}$ .

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 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

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 23

A rectangular hyperbola has double contact with a fixed central conic. If the chord of contact always passes through a fixed point. Prove that the locus of the centre of the hyperbola is a circle passing through the centre of the fixed conic.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 7 - Hyperbola>EXERCISE ON LEVEL-II>Q 25

P

is a variable point on the hyperbola in the form

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

, whose vertex

A

(a, 0)

. Show that the locus of the mid point of

A P

\frac{{(2 x - a)}^{2}}{a^{2}} - \frac{4 y^{2}}{b^{2}} = 1

Find the range of parameter ' a ' for which a unique circle will pass through the point of intersection of the rectangular hyperbola x2-y2=a2 and the parabola y=x2.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

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 $y = x^{2}$ .