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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The length of the common chord of the parabolas $y^{2} = x$ and $x^{2} = y$ and 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

A chord is normal to a parabola

y^{2} = 4 a x

and is inclined at an angle $θ$ to the axis; prove that the area of the triangle formed by it and the tangents at its extremities is

4 a^{2} \sec^{3} θ {cosec}^{3} θ

HARD

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

Two parabolas have the same axis and tangents are drawn to the second from points on the first. Prove that the locus of the middle points of the chords of contact with the second parabola all lie on a fixed 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}

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.

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)

The length of the common chord of the parabolas y2=x and x2=y and is

Important Questions on Parabola

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The length of the common chord of the parabolas $y^{2} = x$ and $x^{2} = y$ and is