1

JEE Main 2018 (Online) 15th April Evening Slot

The tangent to the circle C1 : x2 + y2 $-$ 2x $-$ 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C2 whose center is (3, $-$2). The radius of C2 is :
A
2
B
$\sqrt 2$
C
3
D
$\sqrt 6$

Explanation

Here, equation of tangent on C1 at (2, 1) is :

2x + y $-$ (x + 2) $-$1 = 0

Or    x + y = 3

If it cuts off the chord of the circle C2 then the equation of the chord is :
x + y = 3

$\therefore\,\,\,$ distance of the chord from (3, $-$ 2) is :

d = $\left| {{{3 - 2 - 3} \over {\sqrt 2 }}} \right|$ = $\sqrt 2$

Also, length of the chord is $l$ = 4

$\therefore\,\,\,$ radius of C2 = r = $\sqrt {{{\left( {{l \over 2}} \right)}^2} + {d^2}}$

= $\sqrt {{{\left( 2 \right)}^2} + {{\left( {\sqrt 2 } \right)}^2}} = \sqrt 6$
2

JEE Main 2018 (Online) 16th April Morning Slot

If a circle C, whose radius is 3, touches externally the circle,
${x^2} + {y^2} + 2x - 4y - 4 = 0$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :
A
$2\sqrt 5$
B
$3\sqrt 2$
C
$\sqrt 5$
D
$2\sqrt 3$

Explanation

Given circle is :

x2 + y2 + 2x $-$ 4y $-$4 = 0

$\therefore\,\,\,$ its center is ($-$ 1, 2) and radius is 3 units.

Let A = (x, y) be the center of the circle C

$\therefore$$\,\,\,$ ${{x - 1} \over 2}$ = 2 $\Rightarrow$ x = 5 and ${{y + 2} \over 2}$ = 2 $\Rightarrow$ y = 2

So the center of C is (5, 2) and its radius is 3

$\therefore\,\,\,$ Equation of center C is :

x2 + y2 $-$ 10x $-$ 4y + 20 = 0

$\therefore\,\,\,$ The length of the intercept it cuts on the x-axis

= 2$\sqrt {{g^2} - c} = 2\sqrt {25 - 20} = 2\sqrt 5$
3

JEE Main 2019 (Online) 9th January Morning Slot

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :
A
a, b, c are in A.P.
B
$\sqrt a ,\sqrt b ,\sqrt c$ are in A.P
C
${1 \over {\sqrt b }} + {1 \over {\sqrt c }}$ = ${1 \over {\sqrt a }}$
D
${1 \over {\sqrt b }} = {1 \over {\sqrt a }} + {1 \over {\sqrt c }}$

Explanation

AB = AC + CB

$\sqrt {{{\left( {b + c} \right)}^2} - {{\left( {c - b} \right)}^2}}$ = $\sqrt {{{\left( {b + a} \right)}^2} - {{\left( {b - a} \right)}^2}}$
+ $\sqrt {{{\left( {c + a} \right)}^2} - {{\left( {c - a} \right)}^2}}$

$\Rightarrow$  $\sqrt {2bc}$ = $\sqrt {2ac}$ + $\sqrt {2ab}$

Dividing by $\sqrt {abc}$ we get.

$\Rightarrow$  ${1 \over {\sqrt a }}$ = ${1 \over {\sqrt b }}$ + ${1 \over {\sqrt c }}$
4

JEE Main 2019 (Online) 9th January Evening Slot

If the circles

x2 + y2 $-$ 16x $-$ 20y + 164 = r2

and  (x $-$ 4)2 + (y $-$ 7)2 = 36

intersect at two distinct points, then :
A
r > 11
B
0 < r < 1
C
r = 11
D
1 < r < 11

Explanation

Circles are x2 + y2 $-$ 16x $-$ 20y + 164 = r2 $\Rightarrow$ c1 (8, 10)

and (x $-$ 4)2 + (y $-$ 7)2 = 36

they intersect at two distinct points

$\left| {{r_1} - {r_2}} \right| < {c_1}{c_2} < {r_1} + {r_2}\left\{ {{c_1}{c_2} = \sqrt {16 + 9} = 5} \right\}$

Now  $\left| {r - 6} \right| < 5 < r + 6$

$\left| {r - 6} \right| < 5$

$\Rightarrow$  $- 5 < r - 6 < 5$

$\Rightarrow$  $1 < r < 11\,\,\,\,\,\,\,\,\,...(i)$

$5 < r + 6$

$- 1 < r\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$

from (i) and (ii)

r $\in$ (1, 11)