From any point on the hyperbola x2a2−y2b2=1, tangents are drawn to the hy

Question

From any point on the hyperbola

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

, tangents are drawn to the hyperbola

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

. Then, area cut-off by the chord of contact on the asymptotes is equal to

Options:

A . a/2 sq unit

B . ab sq unit

C . 2ab sq unit

D . 4ab sq unit

Answer: Option D
:
D
Let P

(x_{1}, y_{1})

be a point on the hyperbola

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

The chord of contact of tangents from P to the hyperbola is given by

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

…… (i)
The equation of the asymptotes are

\frac{x}{a} - \frac{y}{b} = 0

and

\frac{x}{a} + \frac{y}{b} = 0

The points of intersection of Equation (i) with the two asymptotes are given by

x_{1} = \frac{2 a}{\frac{x_{1}}{a} + \frac{y_{1}}{b}}, y_{1} = \frac{2 a}{\frac{x_{1}}{a} + \frac{y_{1}}{b}}

x_{2} = \frac{2 a}{\frac{x_{1}}{a} + \frac{y_{1}}{b}}, y_{2} = \frac{2 a}{\frac{x_{1}}{a} + \frac{y_{1}}{b}}

Area of the triangle =

\frac{1}{2} (x_{1} x_{2} - x_{2} y_{1})

= \frac{1}{2} ∣ ∣∣∣ ∣ ⎛ ⎜ ⎝ - \frac{4 a b \times 2}{\frac{x_{1}^{2}}{a^{2}} - \frac{y_{1}^{2}}{b^{2}}} ⎞ ⎟ ⎠ ∣ ∣∣∣ ∣

Was this answer helpful ?

Next Question

More Questions on This Topic :

Question 1. How many tangents to the circle x2+y2=3 are there which are normal to the ellipse x29+y24=1....

Question 2. Consider two points A and B on the ellipse x225+y29=1 circles are drawn having segments of tangents at A and B in b....diameter, then the length of common chord of the circles is

Question 3. If a tangent of slope 2 of the ellipse x2a2+y2b2=1 is normal to the circle x2+y2+4x+1=0 then the maximum value of ab i....circle x2+y2+4x+1=0 then the maximum value of ab is

Question 4. The ratio of the area enclosed by the locus of mid-point of PS and area of the ellipse where P is any point on the ellip....and S is the focus of the ellipse, is

Question 5. A rectangular hyperbola whose centre is C, is cut by any circle of radius r in four points P, Q, R and S. Then, CP2+CQ2+....P, Q, R and S. Then, CP2+CQ2+CR2+CS2 is equal to

Question 6. If (√3)bx+ay=2ab is tangent to the ellipse x2a2+y2b2=1 , then eccentric angle θ is ....

Question 7. The locus of the point which is such that the chord of contact of tangents drawn from it to the ellipse x2a2+y2b2=1 ....a triangle of constant area with the coordinate axes, is

Chapter 1 - GLOBAL STEEL SCENARIO & INDIAN STEEL INDUSTRY Part 1 : (13-04-2024) INDUSTRY AND COMPANY AWARENESS (ICA)

Direction Sense Test Part 1 Reasoning (Hindi)

Chapter 1 - RMHP / OHP / OB & BP Part 1 : (14-02-2024) GPOE

Cube & Cuboid Part 1 Reasoning (Hindi)

Data Interpretation (DI) Basic Concept Reasoning (Hindi)

Counting Figures Part 1 Counting Of Straight Lines Reasoning (Hindi)

Sail E0 free webinar Webinars

Real Numbers Part 7 Class 10 Maths

Real Numbers Part 1 Class 10 Maths

Polynomials Part 1 Class 10 Maths

DSP Combined 1 Free Revision Test SAIL JO E-0

GPOE / GPA Combined 1 Free Revision Test SAIL JO E-0

BSP Plant Specific Review Test SAIL JO E-0

BSL Combined 1 Free Revision Test SAIL JO E-0

GFM Combined 1 Free Revision Test SAIL JO E-0