1

JEE Main 2017 (Online) 8th April Morning Slot

If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles ${\cos ^{ - 1}}\left( {{1 \over 7}} \right)$ and sec$-$1 (7) at the center respectivey, then the distance between these chords, is :
A
${4 \over {\sqrt 7 }}$
B
${8 \over {\sqrt 7 }}$
C
${8 \over 7}$
D
${16 \over 7}$

Explanation

Since cos2$\theta$ = 1/7  $\Rightarrow$ 2 cos2 Q $-$ 1 = 1/7

$\Rightarrow$   2 cos2$\theta$ = 8/7

$\Rightarrow$    cos2 $\theta$ = 4/7

$\Rightarrow$    cos2$\theta$ = ${4 \over 7}$

$\Rightarrow$   cos2$\theta$ = ${2 \over {\sqrt 7 }}$

Also, sec2$\phi$ = 7 = ${1 \over {2{{\cos }^2}\phi - 1}}$ 7

= cos2$\phi$ $-$ 1 = ${1 \over 7}$

= 2 cos2 $\phi$ = ${8 \over 7}$

= cos$\phi$ = ${2 \over {\sqrt 7 }}$

P1P2 = r cos$\theta$ + r cos$\phi$

= ${4 \over {\sqrt 7 }} + {4 \over {\sqrt 7 }}$ = ${8 \over {\sqrt 7 }}$
2

JEE Main 2017 (Online) 8th April Morning Slot

Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is ${3 \over 5}$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilatateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is :
A
8
B
32
C
80
D
40

Explanation

e = 3/5 & 2ae = 6  $\Rightarrow$   a = 5

$\because$   b2 = a2 (1 $-$ e2)

$\Rightarrow$   b2 = 25(1 $-$ 9/25)

$\Rightarrow$   b = 4

$\therefore$   area of required quadrilateral

= 4(1/2 ab) = 2ab = 40
3

JEE Main 2017 (Online) 9th April Morning Slot

If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $\left| c \right|$ is equal to :
A
$2\sqrt 3$
B
$8\sqrt 3$
C
$10\sqrt 3$
D
$16\sqrt 3$

Explanation

c = $-$ 29m $-$ 9m3

a = 2

Given (at2 $-$ a)2 + 4a2t2 = 64

$\Rightarrow$   (a(t2 + 1)) = 8

$\Rightarrow$   t2 + 1 = 4 $\Rightarrow$ t2 = 3

$\Rightarrow$   t = $\sqrt 3$

$\therefore$   c = 2at(2 + t2)

= $2\sqrt 3 \left( 5 \right)$

$\left| c \right|$ = 10$\sqrt 3$
4

JEE Main 2017 (Online) 9th April Morning Slot

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, −1) and (−2, 2) is :
A
${1 \over 2}$
B
${2 \over {\sqrt 5 }}$
C
${{\sqrt 3 } \over 2}$
D
${{\sqrt 3 } \over 4}$

Explanation

Centre at (0, 0)

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

at point (4, $-$ 1)

${{16} \over {{a^2}}} + {1 \over {{b^2}}}$ = 1

$\Rightarrow$   16b2 + a2 = a2b2         . . . .(i)

at point ($-$ 2, 2)

${4 \over {{a^2}}} + {4 \over {{b^2}}} = 1$

$\Rightarrow$   4b2 + 4a2 = a2b2          . . . .(ii)

$\Rightarrow$   16b2 + a2 = 4a2 + 4b2

From equations (i) and (ii)

$\Rightarrow$   3a2 = 12b2

$\Rightarrow$   a2 = 4b2

b2 = a2(1 $-$ e2)

$\Rightarrow$   e2 = ${3 \over 4}$

$\Rightarrow$   e = ${{\sqrt 3 } \over 2}$