1
MCQ (Single Correct Answer)

### JEE Main 2017 (Online) 8th April Morning Slot

Let z$\in$C, the set of complex numbers. Then the equation, 2|z + 3i| $-$ |z $-$ i| = 0 represents :
A
a circle with radius ${8 \over 3}.$
B
a circle with diameter ${{10} \over 3}.$
C
an ellipse with length of major axis ${{16} \over 3}.$
D
an ellipse with length of minor axis ${{16} \over 9}.$

## Explanation

Given,

2 $\,\left| \, \right.$z + 3i$\,\left| \, \right.$ = $\,\left| \, \right.$z $-$i$\,\left| \, \right.$

Let z = x + iy

$\Rightarrow $$\,\,\, 2 \,\left| \, \right. x + iy + 3i \,\left| \, \right. = \,\left| \, \right. x + iy - i \,\left| \, \right. \Rightarrow$$\,\,\,$ 2 $\,\left| \, \right.$ x + i (y + 3)$\,\left| \, \right.$ = $\,\left| \, \right.$ x + i (y $-$ 1)$\,\left| \, \right.$

$\Rightarrow $$\,\,\, 2 \sqrt {{x^2} + {{\left( {y + 3} \right)}^2}} = \sqrt {{x^2} + {{\left( {y - 1} \right)}^2}} \Rightarrow$$\,\,\,$ 4 (x2 + y2 + 6y + 9) = x2 + y2 $-$ 2y + 1

$\Rightarrow $$\,\,\, 3x2 + 3y2 + 26y + 35 = 0 \Rightarrow$$\,\,\,$ x2 + y2 + ${{26} \over 3}$ y + ${{35} \over 3}$ = 0

This is a equation of circle with center ($-$ ${{13} \over 3}$, 0)

$\therefore\,\,\,$ Radius = $\sqrt {0 + {{169} \over 9} - {{35} \over 3}}$

= $\sqrt {{{64} \over 9}}$

= ${8 \over 3}$
2
MCQ (Single Correct Answer)

### JEE Main 2017 (Online) 8th April Morning Slot

The locus of the point of intersection of the straight lines,

tx $-$ 2y $-$ 3t = 0

x $-$ 2ty + 3 = 0 (t $\in$ R), is :
A
an ellipse with eccentricity ${2 \over {\sqrt 5 }}$
B
an ellipse with the length of major axis 6
C
a hyperbola with eccentricity $\sqrt 5$
D
a hyperbola with the length of conjugate axis 3

## Explanation

Here, tx $-$ 2y $-$ 3t = 0  &  x $-$ 2ty + 3 = 0

On solving, we get;

y = ${{6t} \over {2{t^2} - 2}}$ = ${{3t} \over {{t^2} - 1}}$ & x = ${{3{t^2} + 3} \over {{t^2} - 1}}$

Put    t = tan$\theta$

$\therefore$   x = $-$ 3 sec 2$\theta$  &  2y = 3 ($-$ tan 2$\theta$)

$\because$   sec22$\theta$ $-$ tan22$\theta$ = 1

$\Rightarrow$    ${{{x^2}} \over 9}$ $-$ ${{{y^2}} \over {9/4}}$ = 1

which represents at hyperbola

$\therefore$   a2 = 9  &  b2 = 9/4

$\lambda$(T.A.) = 6; e2 = 1 + ${{9/4} \over 9}$ = 1 + ${1 \over 4}$ $\Rightarrow$ e = ${{\sqrt 5 } \over 2}$
3
MCQ (Single Correct Answer)

### JEE Main 2017 (Online) 8th April Morning Slot

If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is :
A
$\sqrt 2 + 1$
B
2(3 + 2 $\sqrt 2$)
C
2($\sqrt 2$ + 1)
D
3 + 2$\sqrt 2$

## Explanation

Tangent to x2 + y2 = 4 is

y = mx $\pm$ 2$\sqrt {1 + {m^2}}$

Also, x2 = 4y

x2 = 4mx + 8$\sqrt {1 + {m^2}}$

or  x2 = 4mx $-$ 8$\sqrt {1 + {m^2}}$

For D = 0

we have; 16m2 + 4.8$\sqrt {1 + {m^2}}$ = 0

$\Rightarrow$   m2 + 2$\sqrt {1 + {m^2}}$ = 0

$\Rightarrow$    m2 = $-$ 2$\sqrt {1 + {m^2}}$

$\Rightarrow$   m4 = 4 + 4m2

$\Rightarrow$   m4 $-$ 4m2 $-$ 4 = 0

$\Rightarrow$    m2 = ${{4 \pm \sqrt {16 + 16} } \over 2}$

$\Rightarrow$   m2 = ${{4 \pm 4\sqrt 2 } \over 2}$

$\Rightarrow$   m2 = 2 + 2$\sqrt 2$
4
MCQ (Single Correct Answer)

### 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$\phi$

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

### EXAM MAP

#### Joint Entrance Examination

JEE Advanced JEE Main

NEET

#### Graduate Aptitude Test in Engineering

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