Prove that the foci of the hyperbola x y = a 2 + b 2 4 are given by x = y = ± a 2 + b 2 2 a .

Let K K ' and L L ' be the asymptotes which are taken as axes and S be the focus having the coordinates S ( x , y ) with respect to these axes. Draw S H parallel to L L ' meeting K K ' in H . If 2 α be the angle between asymptotes, clearly C S will bisect it, and hence ∠ L CS = ∠ S C H = ∠ C S H = α Then C N = N S = 1 2 C S Hence, C H cos ⁡ α = H S cos ⁡ α = 1 2 C S As C H = x and S H = y and C S = a 2 + b 2 , we get : x cos ⁡ α = y cos ⁡ α = 1 2 a 2 + b 2 or x = y = a 2 + b 2 2 cos ⁡ α Also tan ⁡ α = b a ∴ cos ⁡ α = ± a a 2 + b 2 Therefore, x = y = ± a 2 + b 2 2 a Hence proved.

HARD

JEE Advanced

IMPORTANT

Earn 100

Find the asymptotes of the curve $2 x^{2} + 5 x y + 2 y^{2} + 4 x + 5 y = 0$ , and find the general equation of all hyperbolas having the same asymptotes.

Important Questions on The Hyperbola

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 9

Find the equation of the hyperbola, whose asymptotes are the straight lines

x + 2 y + 3 = 0

,and

3 x + 4 y + 5 = 0

, and which passes through the point

(1, - 1)

. Also write down the equation of the conjugate hyperbola.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 10

In a rectangular hyperbola, prove that

C P

and

C D

are equal, and are inclined to the axis at angles which are complementary.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 11

C

is the centre of the hyperbola

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

and the tangent at any point

P

meets the asymptotes in the points

Q

and

R

. Prove that the equation to the locus of the centre of the circle circumscribing the triangle

C Q R

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

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVII>Q 12

A series of hyperbolas are drawn having a common transverse axis of length

2 a

. Prove that the locus of a point

P

on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymptote, is the curve

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

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 1

Prove that the foci of the hyperbola

x y = \frac{a^{2} + b^{2}}{4}

are given by

x = y = \pm \frac{a^{2} + b^{2}}{2 a}

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 2

Show that two concentric hyperbolas, whose axes meet at angle of

45 °

, cut orthogonally.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 3

A straight line always passes through a fixed point, prove that the locus of the middle point of the portion of it, which is intercepted between two given straight lines, is a hyperbola whose asymptotes are parallel to the given lines.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 4

If the ordinate

N P

at any point

P

of an ellipse be produced to

Q,

so that

N Q

is equal to the sub tangent at

P

, prove that the locus of

Q

is hyperbola.

Find the asymptotes of the curve 2x2+5xy+2y2+4x+5y=0, and find the general equation of all hyperbolas having the same asymptotes.

Important Questions on The Hyperbola

Learn and Prepare for any exam you want !

Find the asymptotes of the curve $2 x^{2} + 5 x y + 2 y^{2} + 4 x + 5 y = 0$ , and find the general equation of all hyperbolas having the same asymptotes.