Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 21 :

Three identical circles, each of radius $x$ , are drawn as shown in figure and tangent to a semicircle of radius $a$ , then $x$ in terms of $a$ ?

$\frac{a}{4}$
$\frac{a}{2}$
$\frac{a}{3}$
$\frac{2 a}{3}$

1

2

3

4

(3)

$From the diagram, ∠ A O B = ∠ B O C = ∠ C O D = 60 ° \Rightarrow ∠ Y O D = \frac{π}{6}$

$Let X be the centre of right-hand circle, O X \sin 30 ° = a$

$Now r = O Y = 2 a + a = a = \frac{r}{3}$

Q 22 :

A chord of the circle $x^{2} + y^{2} - 4 x - 6 y = 0$ passing through origin subtends an angle $\tan^{- 1} (\frac{7}{4})$ at the point where the circle meets positive $y$ -axis, then equation of the chord is

$2 x + 3 y = 0$
$x + 2 y = 0$
$x - 2 y = 0$
$2 x - 3 y = 0$

1

2

3

4

(3)

$Equation of tangent at origin is$

$- 2 (x + 0) - 3 (y + 0) = 0 \Rightarrow 2 x + 3 y = 0$

$\tan θ = \frac{7}{4} \Rightarrow$ $| \frac{m + \frac{2}{3}}{1 - \frac{2 m}{3}} |$ $= \frac{7}{4} \Rightarrow \frac{3 m + 2}{3 - 2 m} = \frac{7}{4}$

$\Rightarrow 12 m + 8 = 21 - 14 m \Rightarrow 26 m = 13 \Rightarrow m = \frac{1}{2}$

$∴$ $y = \frac{1}{2} x \Rightarrow x - 2 y = 0$

Q 23 :

Let origin lies inside the circle $x^{2} + y^{2} - x - \sqrt{2} y - c = c > 0$ . A PQ chord through origin (where P,Q lies on circle) is such that $O P = 2, O Q = 8$ (where O is the origin), then the radius of the circle is

5
6
7
8

1

2

3

4

(3)

$O_{1} O = \sqrt{33}$ and $O M = 3$

$\Rightarrow O_{1} M = 2 \sqrt{6}$ and $P M = 5 \Rightarrow O_{1} P = 7$

Q 24 :

If $r_{1}$ and $r_{2}$ are the radii of smallest and largest circles which pass through (5, 6) and touch the circle ${(x - 2)}^{2} + y^{2} = 4$ , then $4 r_{1} r_{2}$ is

$\frac{4}{41}$
$41$
$\frac{5}{41}$
$\frac{41}{6}$

1

2

3

4

(2)

${(x - 2)}^{2} + y^{2} = 4$

$Centre is (2, 0) and radius = 2$

$Distance between (2, 0) and (5, 6) = \sqrt{9 + 36} = 3 \sqrt{5}$

$∴$ $r_{1} r_{2} = \frac{(3 \sqrt{5} - 2) (3 \sqrt{5} + 2)}{4} = \frac{41}{4}$

$∴$ $4 r_{1} r_{2} = 41$

Q 25 :

If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle C, whose center is (3,2), then the diameter of the circle C is:

$\sqrt{20}$
$4$
$12$
$6 \sqrt{2}$

1

2

3

4

(3)

$2 x + 3 y = 12,$ $3 x - 2 y = 5,$ $13 x = 39 \Rightarrow x = 3,$ $y = 2$

Centre $= C = (5, - 2)$ , radius $= r = \sqrt{25 + 4 - 13} = \sqrt{16} = 4$ , CM $= \sqrt{20},$ CP = 6

Q 26 :

Statement-I: If four distinct points (2k, 3k), (1,0), (0,1), (0,0) are concyclic then k = $\frac{5}{13}$ is only the value for k.

Statement-II: There will be two values for k in the above case.

Statement-I is true, Statement-II is true
Statement-I is false, Statement-II is true
Statement-I is true, Statement-II is false
Statement-I is false, Statement-II is false

1

2

3

4

(3)

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

Q 27 :

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x + (k – 1)y + 3 = 0 and $2 x + k^{2} y - 4 = 0$ , If the line x – y + 2 = 0 intersects the circle at the points A and B, then ${(A B)}^{2}$ is equal to: [2026]

10
27
18
34

1

2

3

4

(3)

x + (k – 1)y + 3 = 0

$\Rightarrow Slope, m_{1} = \frac{1}{1 - k}$

$2 x + k^{2} y - 4 = 0 \Rightarrow Slope, m_{2} = \frac{- 2}{k^{2}}$

FIGUR

Now, $m_{1} m_{2} = - 1$

$\Rightarrow (\frac{1}{1 - k}) (\frac{- 2}{k^{2}}) = - 1$

$\Rightarrow (1 - k) (k^{2}) - 2$

$\Rightarrow k^{3} - k^{2} + 2 = 0$

$\Rightarrow (k + 1) [k^{2} - 2 k + 2] = 0$

$\Rightarrow k = - 1$

$∴ x - 2 y + 3 = 0$

2x + y – 4 = 0

After sloving, we get x = 1, y = 2

$∴$ Centre is (1, 2)

Equation of circle is ${(x - 1)}^{2} + {(y - 2)}^{2} = r^{2}$

It passes through (0, 0)

$∴ {(0 - 1)}^{2} + {(0 - 2)}^{2} = r^{2}$

$\Rightarrow r^{2} = 5$

$∴ {(x - 1)}^{2} + {(y - 2)}^{2} = 5$

$\Rightarrow x^{2} + y^{2} - 2 x - 4 y = 0$

and equation of line x – y + 2 = 0

$p = \frac{1 - 2 + 2}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

$A B = 2 \sqrt{r^{2} - p^{2}} = 2 \sqrt{5 - \frac{1}{2}} = 2 (3 / \sqrt{2}) = 3 / \sqrt{2}$

So, ${(A B)}^{2} = 18$

Topic Question Set

Q 21 :

Three identical circles, each of radius $x$ , are drawn as shown in figure and tangent to a semicircle of radius $a$ , then $x$ in terms of $a$ ?

Q 22 :

A chord of the circle $x^{2} + y^{2} - 4 x - 6 y = 0$ passing through origin subtends an angle $\tan^{- 1} (\frac{7}{4})$ at the point where the circle meets positive $y$ -axis, then equation of the chord is

Q 23 :

Let origin lies inside the circle $x^{2} + y^{2} - x - \sqrt{2} y - c = c > 0$ . A PQ chord through origin (where P,Q lies on circle) is such that $O P = 2, O Q = 8$ (where O is the origin), then the radius of the circle is

Q 24 :

If $r_{1}$ and $r_{2}$ are the radii of smallest and largest circles which pass through (5, 6) and touch the circle ${(x - 2)}^{2} + y^{2} = 4$ , then $4 r_{1} r_{2}$ is

Q 25 :

If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle C, whose center is (3,2), then the diameter of the circle C is:

Q 26 :

Statement-I: If four distinct points (2k, 3k), (1,0), (0,1), (0,0) are concyclic then k = $\frac{5}{13}$ is only the value for k.

Statement-II: There will be two values for k in the above case.

Q 27 :

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x + (k – 1)y + 3 = 0 and $2 x + k^{2} y - 4 = 0$ , If the line x – y + 2 = 0 intersects the circle at the points A and B, then ${(A B)}^{2}$ is equal to: [2026]

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Topic Question Set

Q 21 : Three identical circles, each of radius x, are drawn as shown in figure and tangent to a semicircle of radius a, then x in terms of a?

Q 22 : A chord of the circle x2+y2-4x-6y=0 passing through origin subtends an angle tan-1(74) at the point where the circle meets positive y-axis, then equation of the chord is

Q 23 : Let origin lies inside the circle x2+y2-x-2y-c=c>0. A PQ chord through origin (where P,Q lies on circle) is such that OP=2, OQ=8 (where O is the origin), then the radius of the circle is

Q 24 : If r1 and r2 are the radii of smallest and largest circles which pass through (5, 6) and touch the circle (x-2)2+y2=4, then 4r1r2 is

Q 25 : If one of the diameters of the circle x2+y2-10x+4y+13=0 is a chord of another circle C, whose center is (3,2), then the diameter of the circle C is:

Q 26 : Statement-I: If four distinct points (2k, 3k), (1,0), (0,1), (0,0) are concyclic then k = 513 is only the value for k. Statement-II: There will be two values for k in the above case.

Q 27 : Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x + (k – 1)y + 3 = 0 and 2x+k2y–4=0, If the line x – y + 2 = 0 intersects the circle at the points A and B, then (AB)2 is equal to: [2026]

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Q 21 :

Three identical circles, each of radius $x$ , are drawn as shown in figure and tangent to a semicircle of radius $a$ , then $x$ in terms of $a$ ?

Q 22 :

A chord of the circle $x^{2} + y^{2} - 4 x - 6 y = 0$ passing through origin subtends an angle $\tan^{- 1} (\frac{7}{4})$ at the point where the circle meets positive $y$ -axis, then equation of the chord is

Q 23 :

Let origin lies inside the circle $x^{2} + y^{2} - x - \sqrt{2} y - c = c > 0$ . A PQ chord through origin (where P,Q lies on circle) is such that $O P = 2, O Q = 8$ (where O is the origin), then the radius of the circle is

Q 24 :

If $r_{1}$ and $r_{2}$ are the radii of smallest and largest circles which pass through (5, 6) and touch the circle ${(x - 2)}^{2} + y^{2} = 4$ , then $4 r_{1} r_{2}$ is

Q 25 :

If one of the diameters of the circle $x^{2} + y^{2} - 10 x + 4 y + 13 = 0$ is a chord of another circle C, whose center is (3,2), then the diameter of the circle C is:

Q 26 :

Statement-I: If four distinct points (2k, 3k), (1,0), (0,1), (0,0) are concyclic then k = $\frac{5}{13}$ is only the value for k.

Statement-II: There will be two values for k in the above case.

Q 27 :

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x + (k – 1)y + 3 = 0 and $2 x + k^{2} y - 4 = 0$ , If the line x – y + 2 = 0 intersects the circle at the points A and B, then ${(A B)}^{2}$ is equal to: [2026]