A point moves on the given straight line y = m x , prove that the locus of the foot of the perpendicular let fall from the point upon its polar with respect to the ellipse x 2 a 2 + y 2 b 2 = 1 , is a rectangular hyperbola one of whose asymptote is the diameter of the ellipse which is conjugate to the given straight line.

Any point on the given line y = m x , may be taken as h , m h . Its polar with respect to x 2 a 2 + y 2 b 2 = 1 , will be h x a 2 + y m h b 2 = 1 . . . 1 If a line is perpendicular to it, its equation may be written as m h x b 2 - y h a 2 = λ . . . 2 If 2 , passes through h , m h , we get m h 2 b 2 - m h 2 a 2 = λ Substituting for λ in 2 , we get the equation of the line as, m h x b 2 - y h a 2 = h 2 m 1 b 2 - 1 a 2 m x b 2 - y a 2 = h m 1 b 2 - 1 a 2 a 2 x m - b 2 y a 2 b 2 = h m 1 b 2 - 1 a 2 a 2 x m - b 2 y a 2 b 2 = h m a 2 - b 2 a 2 b 2 h m a 2 - b 2 = a 2 x m - b 2 y . . . 3 Again from 1 , we have b 2 x h + a 2 y m h = a 2 b 2 … 4 On dividing 4 by 3 , we get h b 2 x + a 2 y m h m a 2 - b 2 = a 2 b 2 a 2 x m - b 2 y or b 2 y - a 2 m x b 2 x + a 2 m y + m a 2 b 2 a 2 - b 2 = 0 , which is the required locus. Hence, proved.

HARD

JEE Advanced

IMPORTANT

Earn 100

Show that the pole of any tangent to the rectangular hyperbola $x y = c^{2}$ with respect to the circle lies on a concentric and similarly placed rectangular hyperbola.

Important Questions on The Hyperbola

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 27

Prove that the locus of the poles of all normal chord of the rectangular hyperbola

x y = c^{2}

, is the curve

{(x^{2} - y^{2})}^{2} + 4 c^{2} x y = 0

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 28

Prove that the triangle can be inscribed in the hyperbola

x y = c^{2}

, whose sides touch the parabola

y^{2} = 4 a x

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 29

A point moves on the given straight line

y = m x

, prove that the locus of the foot of the perpendicular let fall from the point upon its polar with respect to the ellipse

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

, is a rectangular hyperbola one of whose asymptote is the diameter of the ellipse which is conjugate to the given straight line.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 30

If from a fixed point on a rectangular hyperbola, perpendiculars are let fall on any two conjugate diameters, prove that the straight line joining the feet of these perpendiculars has a constant direction.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 31

A quadrilateral circumscribes a hyperbola, prove that the straight line joining the middle points of its diagonals passing through the center of the curve.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 32

A, B, C

and

D

are the points of intersection of a circle and a rectangular hyperbola. If

A B

passes through the center of the hyperbola, prove that

C D

passes through the center of the circle.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 33

If a circle and rectangular hyperbola meet in four points

P, Q, R

and

S

, show that the orthocenter of the triangles

Q R S, R S P, S P Q

and

P Q R

, also lie on a circle. Prove also that the tangents to the hyperbola at

R

and

S

, meet in a point which lies on the diameter of the hyperbola which is at right angles to

P Q .

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 11 - The Hyperbola>EXAMPLES XXXVIII>Q 34

A series of hyperbolas is drawn, for an asymptote, the principal axes on ellipse. Prove that the common chord of the hyperbola and the ellipse are all parallel to one of the conjugate diameters of the ellipse.

Show that the pole of any tangent to the rectangular hyperbola xy=c2 with respect to the circle lies on a concentric and similarly placed rectangular hyper­bola.

Important Questions on The Hyperbola

Learn and Prepare for any exam you want !

Show that the pole of any tangent to the rectangular hyperbola $x y = c^{2}$ with respect to the circle lies on a concentric and similarly placed rectangular hyperbola.