MEDIUM

JEE Advanced

IMPORTANT

Earn 100

The vertex $A$ of a parabola is joined to any point $P$ on the curve $y^{2} = 4 a x$ and $P Q$ is drawn at right angles to $A P$ to meet the axis in $Q$ . Prove that the projection of $P Q$ on the axis is always equal to the latus rectum.

Important Questions on Conic Sections. The Parabola

MEDIUM

JEE Advanced

IMPORTANT

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

If on a given base(

x

-axis), a triangle is described such that the sum of the tangents of the base angles is constant, prove that the locus of the vertices is a parabola.

HARD

JEE Advanced

IMPORTANT

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

A double ordinate of the curve

y^{2} = 4 p x

is of length

8 p

. Prove that the lines from the vertex to its two ends are at right angles.

HARD

JEE Advanced

IMPORTANT

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

Two parabolas have a common axis and concavities in opposite directions. If any line parallel to the common axis meets the parabolas in

P

and

P^{'}

, prove that the locus of the middle point of

P P^{'}

is another parabola, provided that the latus rectum of the given parabolas are unequal.

MEDIUM

JEE Advanced

IMPORTANT

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

A parabola is drawn to pass through

A

and

B

, the ends of a diameter of a given circle of radius

a

, and to have as directrix a tangent to a concentric circle of radius

b

; the axis being

A B

and a perpendicular diameter, prove that the locus of the focus of the parabola is

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

MEDIUM

JEE Advanced

IMPORTANT

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

The equations of the tangent and the normal at the point

(4, 6)

of the parabola

y^{2} = 9 x

are

c y - 3 x = d & 3 y + a x = b

, respectively. Find

a + b + c + d

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 2

The equations of the tangent and the normal at the point of the parabola

y^{2} = 6 x

, whose ordinate is

12

are

4 y - a x = b & y + c x = d

, respectively. Find

a + b + c + d

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 3

At the ends of the latus rectum of the parabola

y^{2} = 12 x

, the equations of tangents are

x + y = p & x - y = q

and equations of the normals are

x + y = r & x - y = s

, where

p, q, r, s

are integers. Find

p + q + r + s

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 4

At the ends of the latus rectum of the parabola

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

, the equations of tangents are

x + y = p a & x - y = q a

and equations of the normals are

x + y = r a & x - y = s a

, where

p, q, r, s

are integers. Find

p + q + r + s

The vertex A of a parabola is joined to any point P on the curve y2=4ax and PQ is drawn at right angles to AP to meet the axis in Q. Prove that the projection of PQ on the axis is always equal to the latus rectum.

Important Questions on Conic Sections. The Parabola

Learn and Prepare for any exam you want !

The vertex $A$ of a parabola is joined to any point $P$ on the curve $y^{2} = 4 a x$ and $P Q$ is drawn at right angles to $A P$ to meet the axis in $Q$ . Prove that the projection of $P Q$ on the axis is always equal to the latus rectum.