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1

### JEE Main 2013 (Offline)

MCQ (Single Correct Answer)
The circle passing through $$(1, -2)$$ and touching the axis of $$x$$ at $$(3, 0)$$ also passes through the point
A
$$\left( { - 5,\,2} \right)$$
B
$$\left( { 2,\,-5} \right)$$
C
$$\left( { 5,\,-2} \right)$$
D
$$\left( { - 2,\,5} \right)$$

## Explanation

Since circle touches $$x$$-axis at $$(3,0)$$

$$\therefore$$ The equation of circle be

$${\left( {x - 3} \right)^2} + {\left( {y - 0} \right)^2} + \lambda y = 0$$

As it passes through $$(1, -2)$$

$$\therefore$$ Put $$x=1,$$ $$y=-2$$

$$\Rightarrow {\left( {1 - 3} \right)^2} + {\left( { - 2} \right)^2} + \lambda \left( { - 2} \right) = 0$$

$$\Rightarrow \lambda = 4$$

$$\therefore$$ equation of circle is $${\left( {x - 3} \right)^2} + {y^2} - 8 = 0$$

Now, from the options $$\left( {5, - 2} \right)$$ satisfies equation of circle.
2

### AIEEE 2012

MCQ (Single Correct Answer)
The length of the diameter of the circle which touches the $$x$$-axis at the point $$(1, 0)$$ and passes through the point $$(2, 3)$$ is:
A
$${{10} \over 3}$$
B
$${{3} \over 5}$$
C
$${{6} \over 5}$$
D
$${{5} \over 3}$$

## Explanation

Let center of the circle be $$\left( {1,h} \right)$$

$$\left[ {\,\,} \right.$$ as circle touches $$x$$-axis at $$\left. {\left( {1,0} \right)\,\,} \right]$$

Let the circle passes through the point $$B(2,3)$$

$$\therefore$$ $$CA=CB$$ (radius)

$$\Rightarrow C{A^2} = C{B^2}$$

$$\Rightarrow {\left( {1 - 1} \right)^2} + \left( {h - 0} \right){}^2 = {\left( {1 - 2} \right)^2} + {\left( {h - 3} \right)^2}$$

$$\Rightarrow {h^2} = 1 + {h^2} + 9 - 6h$$

$$\Rightarrow h = {{10} \over 6} = {5 \over 3}$$

Thus, diameter is $$2h = {{10} \over 3}.$$
3

### AIEEE 2011

MCQ (Single Correct Answer)
The two circles x2 + y2 = ax, and x2 + y2 = c2 (c > 0) touch each other if
A
| a | = c
B
a = 2c
C
| a | = 2c
D
2 | a | = c

## Explanation

As center of one circle is $$\left( {0,0} \right)$$ and other circle passes through $$(0,0),$$ therefore

Also $${C_1}\left( {{a \over 2},0} \right){C_2}\left( {0,0} \right)$$

$${r_1} = {a \over 2}{r_2} = C$$

$${C_1}{C_2} = {r_1} - {r_2} = {a \over 2}$$

$$\Rightarrow C - {a \over 2} = {a \over 2}$$

$$\Rightarrow C = a$$

If the two circles touch each other, then they must touch each other internally.
4

### AIEEE 2010

MCQ (Single Correct Answer)
The circle $${x^2} + {y^2} = 4x + 8y + 5$$ intersects the line $$3x - 4y - m$$ at two distinct points if
A
$$- 35 < m < 15$$
B
$$15 < m < 65$$
C
$$35 < m < 85$$
D
$$- 85 < m < -35$$

## Explanation

Circle $${x^2} + {y^2} - 4x - 8y - 5 = 0$$

Center $$=(2,4),$$ Radius $$= \sqrt {4 + 16 + 5} = 5$$

If circle is intersecting line $$3x-4y=m,$$ at two distinct points.

$$\Rightarrow$$ length of perpendicular from center to the line $$<$$ radius

$$\Rightarrow {{\left| {6 - 16 - m} \right|} \over 5} < 5 \Rightarrow \left| {10 + m} \right| < 25$$

$$\Rightarrow - 25 < m + 10 < 25 \Rightarrow - 35 < m < 15$$

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