Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

Multiple choice

Let $C$ be the circle with centre $\left( {0,0} \right)$ and radius $3$ units. The equation of the locus of the mid-points of the chords of the circle $C$ that subtend an angles of $\frac{{2\pi }}{3}$ at its centre, is:
  1. ${x^3} + {y^2} = 1$
  2. ${x^2} + {y^2} = \frac{{27}}{4}$
  3. ${x^2} + {y^2} = \frac{9}{4}$
  4. ${x^2} + {y^2} = \frac{3}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a chord of a circle with radius R subtending angle 2*theta at the center, the distance from the center to the chord is d = R * cos(theta). Here R = 3 and 2*theta = 2*pi/3, so theta = pi/3. d = 3 * cos(pi/3) = 3 * 0.5 = 1.5. The locus of the midpoints is a circle with radius d = 1.5, so x^2 + y^2 = (1.5)^2 = 2.25 = 9/4.

Multiple choice

If we can draw three real and distinct chords from $(\alpha, 0), \alpha > 0$ the circle $x^2 + y^2 = a^2$ which are bisected by the parabola $y^2 = 4ax, a > 0$ then
  1. $\alpha = 4a$
  2. $\alpha > 4a$
  3. $\alpha < 4a$
  4. $\alpha = 2a$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a chord of x^2 + y^2 = a^2 to be bisected by y^2 = 4ax at (x1, y1), the midpoint (x1, y1) must satisfy the parabola equation and the chord property. The condition for three distinct chords involves the geometry of the intersection of the circle and the parabola.

Multiple choice

The locus of the middle points of the chords of the circle ${x}^{2}+{y}^{2}=4$ which pass through $(3,4)$ is
  1. ${x}^{2}+{y}^{2}+4x+3y=0$
  2. ${x}^{2}+{y}^{2}-4x-3y=0$
  3. ${x}^{2}+{y}^{2}-3x-4y=0$
  4. ${x}^{2}+{y}^{2}+3x+4y=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the midpoint be (h, k). The chord passing through (3, 4) has a midpoint (h, k). The vector from origin to (h, k) is perpendicular to the chord. The slope of the line from origin to (h, k) is k/h. The slope of the chord is (k-4)/(h-3). Since they are perpendicular, (k/h) * ((k-4)/(h-3)) = -1. k^2 - 4k = -h^2 + 3h => h^2 + k^2 - 3h - 4k = 0. Replacing (h, k) with (x, y) gives x^2 + y^2 - 3x - 4y = 0.

Multiple choice

All chords through an external point to the circle $x^{2}+y^{2}=16$ are drawn having length $\ell$ which is a positive integer. The sum of the squares of the distances from the centre of circle to these chords is
  1. $154$
  2. $124$
  3. $172$
  4. $128$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For any chord of integer length l, the distance d from the center of the circle x^2 + y^2 = 16 (radius R = 4) satisfies d^2 = 16 - l^2/4. Since the chords pass through an external point, for each integer length l from 1 to 7, there are exactly two such chords (tangents to the concentric circle of radius d), while for l = 8 (the diameter), there is only one chord at distance d = 0. Summing the squares of these distances, we get 2 * Sum_{l=1}^{7} (16 - l^2/4) = 224 - 70 = 154.

Multiple choice

Circle $C_1$ has radius $2$ and circle $C_2$ has radius $3$, and the distance between the centers of $C_1$ and $C_2$ is $7$. If two lines, one tangent to both circles and other passing through the center of both circles, intersect at point $P$ which lies between centers of $C_1$ and $C_2$, the the distance between $P$ and the center of $C_1$
  1. $\dfrac{9}{4}\ units$
  2. $\dfrac{7}{3}\ units$
  3. `$\dfrac{8}{3}\ units$
  4. $\dfrac{14}{5}\ units$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The point P divides the distance between centers in the ratio of the radii (2:3) because the tangent line creates similar triangles with the centers. The distance between centers is 7, so the distance from the center of C1 is (2/(2+3)) * 7 = 14/5.

Multiple choice

The length of the common chord of the circle $x ^ { 2 } + y ^ { 2 } + 5 x + 7 y + 9 = 0$ and $x ^ { 2 } + y ^ { 2 } + 7 x + 5 y + 9 = 0$

  1. 1

  2. 2

  3. 4

  4. 6
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The common chord of two circles S1=0 and S2=0 is given by S1 - S2 = 0. (x^2 + y^2 + 5x + 7y + 9) - (x^2 + y^2 + 7x + 5y + 9) = 0 => -2x + 2y = 0 => y = x. Substituting y=x into the first circle: x^2 + x^2 + 5x + 7x + 9 = 0 => 2x^2 + 12x + 9 = 0. This does not yield a standard chord length calculation without further steps. Re-calculating: The circles are x^2+y^2+5x+7y+9=0 and x^2+y^2+7x+5y+9=0. Subtracting gives 2x-2y=0, so y=x. The distance from the center (-2.5, -3.5) to the line x-y=0 is |(-2.5)-(-3.5)|/sqrt(2) = 1/sqrt(2). Radius^2 = 2.5^2 + 3.5^2 - 9 = 6.25 + 12.25 - 9 = 9.5. Chord length = 2 * sqrt(R^2 - d^2) = 2 * sqrt(9.5 - 0.5) = 2 * sqrt(9) = 6.

Multiple choice

The length of the chord joining the points $\left( {4\cos \theta ,4\sin \theta } \right)$ and $\left[ {4\cos \left( {\theta + {{60}^o}} \right),4\sin \left( {\theta + {{60}^o}} \right)} \right]$ of the circle ${x^2} + {y^2} = 16$ is:
  1. $4$
  2. $6$
  3. $2$
  4. $8$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The points are on a circle of radius r=4. The angle between them is 60 degrees. The chord length L = 2 * r * sin(theta/2) = 2 * 4 * sin(60/2) = 8 * sin(30) = 8 * 0.5 = 4.

Multiple choice

The length of the common chord of the circle $x^{2}+y^{2}+2x+3y+1=0$ and $x^{2}+y^{2}+4x+3y+2=0$ is
  1. $\frac{9}{2}$
  2. $2\sqrt{2}$
  3. $3\sqrt{2}$
  4. $\frac{3}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Subtract the two circle equations to find the common chord line: (x^2+y^2+4x+3y+2) - (x^2+y^2+2x+3y+1) = 0, which simplifies to 2x + 1 = 0 or x = -0.5. Substituting x = -0.5 into the first circle equation gives y^2 + 3y + 0.25 = 0, so y = (-3 +/- sqrt(9 - 1))/2 = (-3 +/- 2*sqrt(2))/2. The length of the chord is the difference between these y-values, which is 2*sqrt(2).

Multiple choice

The coordinates of the middle points of the chord cut-off by $2x - y + 18 = 0$ by the circle $x^2 + y^2 - 6x + 2y - 54 = 0$ are
  1. $(-7,4)$
  2. $(2,4)$
  3. $(4,1)$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The midpoint of a chord is the projection of the circle's center onto the line. Center is (3, -1). The line perpendicular to 2x - y + 18 = 0 passing through (3, -1) is x + 2y - 1 = 0. Solving the system 2x - y + 18 = 0 and x + 2y - 1 = 0 gives (-7, 4).