The locus of a variable point whose chord of contact w.r.t. the hyperbola x 2 a 2 - y 2 b 2 = 1 subtends a right angle at the origin is

x 2 a 4 + y 2 b 4 = 1 a 2 - 1 b 2

x 2 a 2 - y 2 b 2 = x 2 a 4 + y 2 b 4

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Equation of the chord of the hyperbola $25 x^{2} - 16 y^{2} = 400$ which is bisected at the point (6, 2), is

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Important Questions on Hyperbola

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Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

Tangents are drawn to the hyperbola

4 x^{2} - y^{2} = 36

at the points

P

and

Q

. If these tangents intersect at the point

T (0, 3)

then the area (in sq. units) of

Δ P T Q

is:

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

If any tangent to the parabola

x^{2} = 4 y

intersects the hyperbola

x y = 2

at two points

P

and

Q,

then the mid-point of line segment

P Q

lies on a parabola with axis along

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of a variable point whose chord of contact w.r.t. the hyperbola

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

subtends a right angle at the origin is

EASY

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

If the pole of the line

3 x - 16 y + 48 = 0

with respect to the hyperbola

9 x^{2} - 16 y^{2} = 144

(α, β),

then

α - β =

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The lines

x \cos α + y \sin α = P, α \in ℝ

are chords of the hyperbola

\frac{x^{2}}{9} - \frac{y^{2}}{36} = 1

and they subtend a right angle at the centre of the hyperbola. The locus of the poles of these lines with respect to the given hyperbola is

EASY

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The equation of the chord of the hyperbola

25 x^{2} - 16 y^{2} = 400

that is bisected at point

(5, 3)

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of middle points of chords of hyperbola

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

parallel to

y = 2 x

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

If the chords of the hyperbola

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

touch the parabola

y^{2} = 4 a x

. Then, the locus of the middle points of these chords is

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The equation of the chord of the hyperbola

25 x^{2} - 16 y^{2} = 400

, which is bisected at the point

(6, 2),

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of middle points of chords of hyperbola

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

parallel to

y = 2 x

EASY

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

x = 9

is the chord of contact of the hyperbola

x^{2} - y^{2} = 9

, then the equation of the corresponding pair of tangent is

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of middle points of chords of hyperbola

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

parallel to

y = 2 x

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of middle points of chords of hyperbola

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

parallel to

y = 2 x

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

From any point on the hyperbola

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

tangents are drawn to the hyperbola

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

The area cut off by the chord of contact on the region between the asymptotes is equal to -

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

Chords of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ are tangents to the circle drawn on the line joining the foci as diameter. Find the locus of the point of intersection of tangents at extremities of the chords.

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

If the chords of contact of tangents from two points (x₁, y₁) and (x₂, y₂) to the hyperbola 4x² - 9y² - 36 = 0 are at right angles, then $\frac{x_{1} x_{2}}{y_{1} y_{2}}$ is equal to

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

If the chords of contact of tangents from two points

(x_{1}, y_{1})

amd

(x_{2}, y_{2})

to the hyperbola

4 x^{2} - 9 y^{2} - 36 = 0

are at right angles, then

\frac{x_{1} x_{2}}{y_{1} y_{2}}

equal to

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of middle points of chords of hyperbola

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

parallel to

y = 2 x

HARD

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The locus of middle points of chords of hyperbola

3 x^{2} - 2 y^{2} + 4 x - 6 y = 0

parallel to

y = 2 x

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Pair of Tangents, Chord of Contact and Chord with Midpoint of a Hyperbola

The mid point of the chord

4 x - 3 y = 5

of the hyperbola

2 x^{2} - 3 y^{2} = 12

Equation of the chord of the hyperbola 25x2-16y2=400 which is bisected at the point (6, 2), is

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

Equation of the chord of the hyperbola $25 x^{2} - 16 y^{2} = 400$ which is bisected at the point (6, 2), is