The axes being rectangular, prove that the locus of the focus of the parabola x a + y b - 1 2 = 4 x y a b , a and b being variables such that a b = c 2 , is the curve x 2 + y 2 2 = c 2 x y .

Given equation of parabola is x a + y b - 1 2 = 4 x y a b i.e. ( b x + a y - a b ) 2 = 4 a b x y or b 2 x 2 - 2 a b x y + a 2 y 2 - 2 a b 2 x - 2 a 2 b y + a 2 b 2 = 0 Let coordinates of focus be ( h , k ) . Now we know that algebraically, equation of pair of tangents drawn from focus to its conic satisfy the analytical conditions of a circle. Now, equation of pair of tangents is T 2 = S S 1 i.e. b 2 x h - a b ( x k + h y ) + a 2 y k - a b 2 ( x + h ) - a 2 b ( y + k ) + a 2 b 2 2 = ( b 2 x 2 - 2 a b x y + a 2 y 2 - 2 a b 2 x - 2 a 2 b y + a 2 b 2 ) S 1 ⇒ b ( b h - a k - a b ) x - a ( b h - a k + a b ) y - a b ( b h + a k - a b ) 2 = b 2 S 1 x 2 + a 2 S 1 y 2 - 2 a b S 1 x y + g ( x , y ) ⇒ b 2 ( b h - a k - a b ) 2 x 2 - 2 a b ( b h - a k - a b ) ( b h - a k + a b ) x y + a 2 ( b h - a k + a b ) 2 y 2 + f ( x , y ) = b 2 S 1 x 2 + a 2 S 1 y 2 - 2 a b S 1 x y + g ( x , y ) ⇒ b 2 ( b h - a k - a b ) 2 - S 1 x 2 + a 2 ( b h - a k + a b ) 2 - S 1 y 2 - 2 a b ( b h - a k - a b ) ( b h - a k + a b ) - S 1 x y + f ( x , y ) - g ( x , y ) = 0 . . . ( 1 ) Let ( b h - a k - a b ) = A and ( b h - a k + a b ) = B Since ( 1 ) satisfies analytical condition for circle i.e. coefficient of x 2 = coefficient of y 2 and coefficient of x y = 0 . Thus, b 2 ( A 2 - S 1 ) = a 2 ( B 2 - S 1 ) . . . ( 2 ) and A B - S 1 = 0 i.e. S 1 = A B . . . ( 3 ) Putting ( 3 ) in ( 2 ) , we get b 2 A = - a 2 B ⇒ b 2 ( b h - a k - a b ) = - a 2 ( b h - a k + a b ) ⇒ ( b 2 + a 2 ) ( b h - a k ) = a b 3 - a 3 b = ( a b ) ( b 2 - a 2 ) = ( c 2 ) ( b 2 - a 2 ) ( ∵ a b = c 2 is given ) ⇒ b h - a k = b 2 - a 2 b 2 + a 2 c 2 . . . ( 4 ) From ( 3 ) , S 1 = A B ⇒ b 2 h 2 - 2 a b h k + a 2 k 2 - 2 a b 2 h - 2 a 2 b k + a 2 b 2 = ( b h - a k - a b ) ( b h - a k + a b ) ⇒ b h + a k - a b = 0 ⇒ b h + a k = c 2 . . . ( 5 ) Solving ( 4 ) and ( 5 ) , we get h = b c 2 b 2 + a 2 and k = a c 2 b 2 + a 2 . . . ( 6 ) To find required locus we need to eliminate a and b from ( 6 ) . Thus we get, h 2 + k 2 2 = c 2 h k Generalising the locus, we get x 2 + y 2 2 = c 2 x y Hence, Proved.

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Equation of chord of the parabola $y^{2} = 16 x$ whose mid point is (1, 1), is

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Important Questions on Parabola

MEDIUM

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

Let

P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)

be three points on a parabola

y^{2} = 4 a x

. If

P Q

is the focal chord and

P K, Q R

are parallel where the co-ordinates of

K

(2 a, 0),

then the value of

r

MEDIUM

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

Let

P : y^{2} = 4 a x, a > 0

be a parabola with focus

S

.Let the tangents to the parabola

P

make an angle of

\frac{π}{4}

with the line

y = 3 x + 5

touch the parabola

P

A

and

B

. Then the value of

a

for which

A, B

and

S

are collinear is:

HARD

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

The tangents to the curve

y = {(x - 2)}^{2} - 1

at its points of intersection with the line

x - y = 3,

intersect at the point:

MEDIUM

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

Find the equation of that chord of parabola

y^{2} = 16 x

, whose middle point is

(4, 0)

HARD

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

If a chord, which is not a tangent, of the parabola

y^{2} = 16 x

has the equation

2 x + y = p

, and midpoint

(h, k)

, then which of the following is (are) possible value(s) of

p, h

and

k

HARD

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

Find the locus of the middle points of chords of the parabola

y^{2} = 4 a x

which are of given length

l .

HARD

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

Two equal parabolas with axis in opposite directions touch at a point

O

. From a point

P

on one of them, tangents

P Q

and

P Q^{'}

are drawn to the other. Prove that,

Q Q^{'}

will touch the first parabola in

P^{'}

where,

P P^{'}

is parallel to the common tangent at

O

HARD

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

Find the locus of the middle points of chords of the parabola which are such that the normal at their extremities meet on the parabola.

HARD

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

Through each point of the straight line

x = m y + h

a chord of the parabola

y^{2} = 4 a x

is drawn, which is bisected at the point. Prove that it always touches the parabola

{(y + 2 a m)}^{2} = 8 a (x - h)

HARD

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

Locus of midpoint of any focal chord of

y^{2} = 4 a x

EASY

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

P

is a point on the parabola

y^{2} = 8 x

and

A

is the point

(1, 0)

, then the locus of the midpoint of the line segment

A P

HARD

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

For the parabola

y^{2} = 4 a x

, prove that the locus of the poles of chords which subtend a right angle at a fixed point

(h, k)

a x^{2} - h y^{2} + (4 a^{2} + 2 a h) x - 2 a k y + a (h^{2} + k^{2}) = 0 .

HARD

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

Find the locus of the middle points of the chords of the parabola which are normal to the curve.

MEDIUM

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

What is the equation to the chord of the parabola

y^{2} = 8 x

which is bisected at the point

(2, - 3)

HARD

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

Show that the locus of the poles of tangents to the parabola

y^{2} = 4 a x

with respect to the parabola

y^{2} = 4 b x

is the parabola

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

MEDIUM

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

Two tangents are drawn from a point

(- 2, - 1)

to the curve

y^{2} = 4 x

. If

α

is the angle between them, then

|\tan α|

is equal to

HARD

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

The axes being rectangular, prove that the locus of the focus of the parabola

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

a

and

b

being variables such that

a b = c^{2}

, is the curve

{(x^{2} + y^{2})}^{2} = c^{2} x y

HARD

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

For parabola

y^{2} = 4 a x, a > 0

. Prove that the length of the chord joining the points of contact of the tangents drawn from the point

(x_{1}, y_{1})

\frac{\sqrt{y_{1}^{2} + 4 a^{2}} \sqrt{y_{1}^{2} - 4 a x_{1}}}{a}

MEDIUM

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

Find the locus of the middle point of chord of the parabola

y^{2} = 4 a x

which passes through the fixed point

(h, k)

HARD

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

Find the locus of the middle points of the chords of the parabola which subtend a constant angle

θ

at the vertex.

Equation of chord of the parabola y2=16x whose mid point is (1, 1), is

Important Questions on Parabola

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Equation of chord of the parabola $y^{2} = 16 x$ whose mid point is (1, 1), is