MEDIUM

JEE Advanced

IMPORTANT

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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.

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 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

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 5

The equation of that tangent to the parabola

y^{2} = 7 x

which is parallel to the straight line

4 y - x + 3 = 0

4 y - x = a

and its point of contact is

(b, c)

. Find

a - b + c

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 6

A tangent to the parabola

y^{2} = 4 a x

makes an angle of

60 °

with the axis and its point of contact is

(\frac{a}{3}, \frac{k a}{\sqrt{3}})

, then find

k

EASY

JEE Advanced

IMPORTANT

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

The points on the parabola

y^{2} = 4 a x

at which

(i)

the tangent and

(i i)

the normal is inclined at

30 °

to the axis of the parabola are

(3 a, 2 \sqrt{β} a)

and

(\frac{a}{3}, - \frac{2 \sqrt{γ}}{3} a)

, respectively. Find

β + γ

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 10

Prove that the straight line

x + y = 1

touches the parabola

y = x - x^{2}

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVI>Q 11

Prove that the straight line

y = m x + c

touches the parabola

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

, if

c = m a + \frac{a}{m}

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.

Important Questions on Conic Sections. The Parabola

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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.