Ellipse And Hyperbola(12th Grade > Mathematics ) Questions and answers for exam Preparation

12th Grade > Mathematics

ELLIPSE AND HYPERBOLA MCQs

Total Questions : 30 | Page 3 of 3 pages

Question 21. If PSQ is a focal chord of the ellipse

16 x^{2} + 25 y^{2} = 400

such that SP = 8 then the length of SQ =

Discuss Question

Answer: Option A. -> 2
:
A

\frac{1}{S P} + \frac{1}{S Q} = \frac{2 a}{b^{2}}

a= 5
b=4
Solving we get,
SQ=2.

Question 22. If x = 9 is the chord of contact of the hyperbola

x^{2} - y^{2} = 9

, then the equation of the corresponding pair of tangents is

9x2−8y2+18x−9=0
9x2−8y2−18x+9=0
9x2−8y2−18x−9=0
9x2−8y2+18x+9=0

Discuss Question

Answer: Option B. -> 9x2−8y2−18x+9=0
:
B
If x = 9 meets the hyperbola at

(9, 6 \sqrt{2})

and

(9, - 6 \sqrt{2})

.
Then, equations of tangent at these points are

3 x - 2 \sqrt{2} y - 3 = 0

and

3 x + 2 \sqrt{2} y - 3 = 0

.
The combined equation of these two is

9 x^{2} - 8 y^{2} - 18 x + 9 = 0

Question 23. 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

a/2 sq unit
ab sq unit
2ab sq unit
4ab sq unit

Discuss Question

Answer: Option D. -> 4ab sq unit
:
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}}} ⎞ ⎟ ⎠ ∣ ∣∣∣ ∣

Question 24. How many tangents to the circle

x^{2} + y^{2} = 3

are there which are normal to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

Discuss Question

Answer: Option D. -> 0
:
D
Equation of normal at

p (3 c o s θ, 2 s i n θ)

3 x s e c θ - 2 y c o s e c θ = 5

\frac{5}{\sqrt{9 s e c^{2} θ + 4 c o s e c^{2} θ}} = \sqrt{3} B u t m i n i m u m v a l u e o f 9 s e c^{2} θ + 4 c o s e c^{2} θ = 25 ∴ n o s u c h θ^{- 1} e x i s t s

So the number of tangents to the circle

x^{2} + y^{2} = 3

which are normal to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

is 0.

Question 25. Consider two points A and B on the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1

circles are drawn having segments of tangents at A and B in between tangents at the two ends of major axis of ellipse as diameter, then the length of common chord of the circles is

8
6
10
4√2

Discuss Question

Answer: Option A. -> 8
:
A
All such circles pass through foci

∴

The common chord is of the length 2ae =

10 \times \frac{4}{5} = 8

Question 26. If a tangent of slope 2 of the ellipse

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

is normal to the circle

x^{2} + y^{2} + 4 x + 1 = 0

then the maximum value of ab is

2
4
6
Can’t be found

Discuss Question

Answer: Option B. -> 4
:
B
A tangent of slope 2 is

y = 2 x \pm \sqrt{4 a^{2} + b^{2}}

,this is normal to

x^{2} + y^{2} + 4 x + 1 = 0

then

0 = - 4 \pm \sqrt{4 a^{2} + b^{2}} \Rightarrow 4 a^{2} + b^{2} = 16

u s i n g A . M . \geq G . M ., W e g e t a b \leq 4

Question 27. 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 ellipse and S is the focus of the ellipse, is

Discuss Question

Answer: Option D. -> 14
:
D