Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 5

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If $θ_{1}, θ_{2}, θ_{3}, θ_{4}$ be eccentric angles of the four concyclic points of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , then $θ_{1} + θ_{2} + θ_{3} + θ_{4}$ is $($ where $n \in I)$

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Important Questions on Conic Section

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 8

Area of the region bounded by the curve

\{(x, y) : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \leq 1 \leq \frac{x}{a} + \frac{y}{b}\}

is

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 9

The ellipse

x^{2} + 4 y^{2} = 4

is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point

(4, 0)

. Then the equation of the ellipse is

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 10

If

P Q

is a double ordinate of the hyperbola

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

, such that

O P Q

is an equilateral triangle,

O

being the centre of the hyperbola, then the eccentricity

e

of the hyperbola satisfies

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 11

A normal to the parabola

y^{2} = 4 p x

with slope

m

touches the rectangular hyperbola

x^{2} - y^{2} = p^{2}

, if

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 12

The locus of a point

P (α, β)

, such that the line

y = α x + β

is tangent to

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

, is

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 13

Let

P (a \sec θ, b \tan θ)

and

Q (a \sec ϕ, b \tan ϕ)

, where

θ + ϕ = \frac{π}{2}

, be two points on the hyperbola

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

. If

(h, k)

is the point of intersection of the normals at

P

and

Q

, then

k

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 14

An ellipse has eccentricity

\frac{1}{2}

and one focus at the point

P (\frac{1}{2}, 1) .

Its one directrix is the common tangent nearer to the point

P,

to the circle

x^{2} + y^{2} = 1

and the hyperbola

x^{2} - y^{2} = 1 .

The equation of the ellipse in the standard form is

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Gamma Question Bank for Engineering Mathematics>Chapter 14 - Conic Section>Assignment>Q 15

If the polars of

(x_{1}, y_{1})

and

(x_{2}, y_{2})

w.r.t. the hyperbola

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

are at right angles, then the value of

\frac{x_{1} x_{2}}{y_{1} y_{2}}

is