Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 5

MEDIUM

JEE Main/Advanced

IMPORTANT

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For parabola $x^{2} + y^{2} + 2 x y - 6 x - 2 y + 3 = 0$ , the focus is :

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Important Questions on Parabola

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 6

A ray of light moving parallel to the

x

-axis gets reflected from a parabolic mirror whose equation is

{(y - 2)}^{2} = 4 (x + 1)

. After reflection, the ray must pass through the point:

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 1

Prove that the locus of the point of intersection of tangents to the parabola $y^{2} = 4 a x$ which meet at an angle $α$ is

${(x + a)}^{2} \tan^{2} α = y^{2} - 4 a x$ .

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 2

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

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 3

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 .

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 4

Show that the polar of any point on the circle

x^{2} + y^{2} - 2 a x - 3 a^{2} = 0

with respect to the circle

x^{2} + y^{2} + 2 a x - 3 a^{2} = 0

will touch the parabola

y^{2} + 4 a x = 0

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>INTRODUCTORY EXERCISE 5.3>Q 5

From the point

P (- 1, 2)

tangents are drawn to the parabola

y^{2} = 4 x

. Find the equation of the chord of contact. Also, find the area of the triangle formed by the chord of contact and the tangents.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 1

If the normal at

P (18, 12)

to the parabola

y^{2} = 8 x

cuts it again at

Q

, show that

9 P Q = 8 \sqrt{109}

.

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 5 - Parabola>EXERCISE ON LEVEL-I>Q 2

Show that the length of the chord of the parabola

y^{2} = 4 a x

passing through the vertex and making an angle

θ

with the axis is

\frac{4 a \cos θ}{\sin^{2} θ}

.