Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 64

HARD

KVPY Aptitude Test - Stream SA

IMPORTANT

Earn 100

If a variable line has its intercepts on the coordinate axes as $e and e',$ where $\frac{e}{2}, \frac{e'}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle $x^{2} + y^{2} = r^{2},$ where $r =$

40% studentsanswered this correctly

Important Questions on Conic Section

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 65

The equation of the transverse axis of the hyperbola

(x - 3)^{2} + (y + 1)^{2} = (4 x + 3 y)^{2}

is

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 66

The equation

16 x^{2} - 3 y^{2} - 32 x + 12 y - 44 = 0

represents a hyperbola

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 67

Let

L L^{'}

be the latus rectum through the focus of the hyperbola

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

and

A

be the farther vertex. If

{ΔALL}^{'}

is equilateral, then the eccentricity of the hyperbola is (axes are coordinate axes)

EASY

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 68

The equations of the transverse and conjugate axes of a hyperbola are respectively

x + 2 y - 3 = 0, 2 x - y + 4 = 0,

and their respective lengths are

\sqrt{2}

and

\frac{2}{\sqrt{3}} .

The equation of the hyperbola is

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 69

If the eccentricity of the hyperbola

x^{2} - y^{2} \sec^{2} α = 5

is

\sqrt{3}

times the eccentricity of the ellipse

x^{2} \sec^{2} α + y^{2} = 25,

then a value of

α

is

HARD

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 70

The sides

A C

and

A B

of

Δ A B C

touch the conjugate hyperbola of the hyperbola

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

If the vertex

A

lies on the ellipse

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

then the side

B C

must touch

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 71

Portion of asymptote of hyperbola

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

(between centre and the tangent at vetex) in the first quadrant is cut by the line

y + λ (x - a) = 0 (λ

is a parameter

)

then

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 12 - Conic Section>Exercise>Q 72

The locus of the foot of the perpendicular from the centre of the hyperbola

x y = 1

on a variable tangent is