NEW
New Website Launch
Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...
1

WB JEE 2009

MCQ (Single Correct Answer)

The angle between the line joining the foci of an ellipse to one particular extremity of the minor axis is 90$$^\circ$$. The eccentricity of the ellipse is

A
$${1 \over 8}$$
B
$${1 \over {\sqrt 3 }}$$
C
$$\sqrt {{2 \over 3}} $$
D
$$\sqrt {{1 \over 2}} $$

Explanation

Let the equation of the ellipse is $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

coordinates of foci are (ae, 0), ($$-$$ae, 0)

coordinates of extremity of minor axis are (0, b)

Slopes of lines joining foci to extremity are $$ - {b \over {ae}},{b \over {ae}}$$

but lines are at right angle

$$\therefore$$ $$ - {b \over {ae}}.{b \over {ae}} = - 1 \Rightarrow {b^2} = {a^2}{e^2}$$ .... (i)

But $${b^2} = {a^2}(1 - {e^2})$$

$$ \Rightarrow {a^2}{e^2} = {a^2}(1 - {e^2}) \Rightarrow 2{e^2} = 1$$

$$ \Rightarrow e = {1 \over {\sqrt 2 }}$$

2

WB JEE 2009

MCQ (Single Correct Answer)

The line y = 2t2 intersects the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ in real point if

A
$$|t| \le 1$$
B
$$|t| < 1$$
C
$$|t| > 1$$
D
$$|t| \ge 1$$

Explanation

Keep y = 2t2 in $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$, we get

$${{{x^2}} \over 9} + {{{{(2{t^2})}^2}} \over 4} = 1 \Rightarrow {x^2} = 9(1 - {t^4})$$

If line intersects the ellipse in real points, then x has real roots

So $${x^2} \ge 0 \Rightarrow 9(1 - {t^4}) \ge 0 \Rightarrow (1 - {t^2})(1 + {t^2}) \ge 0$$

$$ \Rightarrow 1 - {t^2} \ge 0 \Rightarrow |t| \ge 1$$ [$$\because$$ $$1 + {t^2} > 0$$]

3

WB JEE 2009

MCQ (Single Correct Answer)

The total number of tangents through the point (3, 5) that can be drawn to the ellipse 3x2 + 5y2 = 32 and 25x2 + 9y2 = 450 is

A
0
B
2
C
3
D
4

Explanation

$$\because$$ 3 . 32 + 5 . 52 $$-$$ 32 > 0

$$\therefore$$ From (3, 5) we can draw 2 tangents to the ellipse

3x2 + 5y2 = 32. Again 25 . 32 + 9 . 52 $$-$$ 450 = 0

$$\therefore$$ From (3, 5) we can draw only 1 tangent to the ellipse

25x2 + 9y2 = 450 $$\therefore$$ Required number of tangents = 2 + 1 = 3

Joint Entrance Examination

JEE Main JEE Advanced WB JEE

Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN

Medical

NEET

CBSE

Class 12