Prove that the chord of the parabola y 2 = 4 a x , whose equation is y - x 2 + 4 a 2 = 0 , is a normal to the curve and its length is 6 3 a .

The equation to the parabola is given as y 2 = 4 a x . . . ( i ) and the equation to the chord is y - x 2 + 4 a 2 = 0 . . . i i and its slope m = 2 Now, equation of normal of y 2 = 4 a x of slope 2 , is y = m x - 2 a m - a m 3 ⇒ y = 2 x - 2 a 2 - a 2 2 ⇒ y - x 2 + 4 a 2 = 0 Hence, we can say that given chord is also the normal of the parabola. Now, solve the equations i and i i simultaneously, we put the value of y from i i in i , we get x 2 - 4 a 2 2 = 4 a x 2 x 2 + 32 a 2 - 16 a x = 4 a x x 2 - 10 a x + 16 a 2 = 0 Where, either x = 8 a or x = 2 a Putting in equation i i , the corresponding values of y are 4 a 2 and - 2 a 2 , respectively. Hence, the points of intersection of the equations i and i i are 8 a , 4 a 2 and 2 a , - 2 a 2 . Distance between the points 8 a , 4 a 2 and 2 a , - 2 a 2 = 8 a - 2 a 2 + 4 a 2 + 2 a 2 2 = 36 a 2 + 72 a 2 = 6 a 3 Hence, length of the chord = 6 a 3

HARD

JEE Advanced

IMPORTANT

Earn 100

Find the equation to the common tangents of the parabolas $y^{2} = 4 a x$ and $x^{2} = 4 b y$ .

Important Questions on Conic Sections. The Parabola

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 15

Find the equations to the common tangents of the circle $x^{2} + y^{2} = 4 a x$ and the parabola $y^{2} = 4 a x$ .

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 16

Two equal parabolas have the same vertex and their axes are at right angles. Prove that the common tangent touches each at the end of a latus rectum.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 17

Prove that two straight lines, one a tangent to the parabola

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

and the other to the parabola

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

, which are at right angles to one another meet on the straight line

x + a + a^{'} = 0

. Show also that this straight line is the common chord of the two parabolas.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 18

P N

is an ordinate of the parabola. A straight line is drawn parallel to its axis that bisects

N P

and it meets the curve at

Q

. Prove that

N Q

meets the tangent at the vertex at a point

T

, such that

A T = (\frac{2}{3}) N P

, where

A

is the vertex of the parabola.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 19

Prove that the chord of the parabola

y^{2} = 4 a x

, whose equation is

y - x \sqrt{2} + 4 a \sqrt{2} = 0

, is a normal to the curve and its length is

6 \sqrt{3} a

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 20

If the perpendiculars are drawn on any tangent to a parabola from two fixed points on the axis, which are equidistant from the focus, prove that the difference of their squares is constant.

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 21

P, Q

and

R

be three points on a parabola whose ordinates are in geometrical progression, prove that the tangents at

P

and

R

meet on the ordinate of

Q

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 22

Tangents are drawn to a parabola at points whose abscissae are in the ratio

μ : 1

. Prove that they intersect on the curve

y^{2} = {[μ^{\frac{1}{4}} + μ^{- \frac{1}{4}}]}^{2} a x

Find the equation to the common tangents of the parabolas y2=4ax and x2=4by.

Important Questions on Conic Sections. The Parabola

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Find the equation to the common tangents of the parabolas $y^{2} = 4 a x$ and $x^{2} = 4 b y$ .