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

If the lengths of the chords intercepted by the circle ${x}^{2}+{y}^{2}+2gx+2fy=0$ from the coordinate axes are $10$ and $24$ units, respectively, then the radius of the circle is
  1. $17$
  2. $9$
  3. $14$
  4. $13$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The length of the intercept on the x-axis is 2 * sqrt(g^2 - c) = 10, so g^2 - c = 25. The length of the intercept on the y-axis is 2 * sqrt(f^2 - c) = 24, so f^2 - c = 144. For a circle passing through the origin (as given by the equation), c=0. Thus g^2 = 25 and f^2 = 144. Radius = sqrt(g^2 + f^2 - c) = sqrt(25 + 144) = sqrt(169) = 13.

Multiple choice

Two distinct chords drawn from the point $(p, q)$ on the circle $x^{2}+y^{2}=px+qy$, where $pq\neq 0$, are bisected by $x$-axis, Then
  1. $\left | p \right |=\left | q \right |$
  2. $p^{2}=8q^{2}$
  3. $p^{2}< 8q^{2}$
  4. $p^{2}> 8q^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A chord of x^2 + y^2 - px - qy = 0 bisected by the x-axis (y=0) has a midpoint (h, 0). The equation of the chord with midpoint (h, 0) is T = S1. This leads to the condition p^2 > 8q^2 for two distinct chords to exist.

Multiple choice

Two parallel chords are drawn on the same side of the centre of a circle of radius 20. It is found that they subtend ${ 60 }^{ 0 }$ and ${ 120 }^{ 0 }$ angles at the centre of the circle. Then the perpendicular distance between the chords is:
  1. $5(\sqrt { 3 } -1)$
  2. $10(\sqrt { 3 } -1)$
  3. $10(\sqrt { 2 } -1)$
  4. $5(\sqrt { 3 } +1)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distance from the center to a chord is d = R * cos(theta/2). For the 60 degree angle, the distance is 20 * cos(30) = 20 * (sqrt(3)/2) = 10 * sqrt(3). For the 120 degree angle, the distance is 20 * cos(60) = 20 * (1/2) = 10. Since both chords are on the same side, the perpendicular distance between them is 10 * sqrt(3) - 10 = 10 * (sqrt(3) - 1).

Multiple choice

The chord of circle $C$, having centre at $(2,2)$ is the diameter of the circle $x^{2}+y^{2}-4x+6y-3=0$. If $r$ is radius of the circle $C$, then $r^{2}$ is equal to
  1. $41$
  2. $31$
  3. $51$
  4. $61$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circle x^2 + y^2 - 4x + 6y - 3 = 0 has center (2, -3) and radius sqrt(2^2 + (-3)^2 - (-3)) = sqrt(4 + 9 + 3) = 4. The chord of circle C is the diameter of this circle, so the chord length is 2 * 4 = 8. The distance from C(2, 2) to the chord (which is the line passing through (2, -3) perpendicular to the chord) is 5. Using r^2 = (chord/2)^2 + d^2, r^2 = 4^2 + 5^2 = 16 + 25 = 41.

Multiple choice

In a circle of radius $25$ cm two parallel chords of the length $14$ cm and $48$ cm respectively, are drawn on the same side of the centre. The distance between them is
  1. $14$ cm
  2. $24$ cm
  3. $17$ cm
  4. $31$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a circle of radius 25, the distance from the center to a chord of length L is sqrt(R^2 - (L/2)^2). For L=14, d1 = sqrt(25^2 - 7^2) = sqrt(625 - 49) = 24. For L=48, d2 = sqrt(25^2 - 24^2) = sqrt(625 - 576) = 7. Since they are on the same side, the distance between them is 24 - 7 = 17 cm.

Multiple choice

$HI $ and $JK$ are two parallel chords of a circle with centre $O$. They are opposite sides to the centre such that $HI = 8$ cm, $JK = 6$ cm and the distance between them is $7$ cm. Then, the radius of the circle is
  1. $10$ cm
  2. $4$ cm
  3. $5$ cm
  4. $3$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let r be the radius. Distance from center to chord HI (length 8) is sqrt(r^2 - 4^2). Distance to chord JK (length 6) is sqrt(r^2 - 3^2). Since they are on opposite sides, sqrt(r^2-16) + sqrt(r^2-9) = 7. Testing r=5: sqrt(25-16) + sqrt(25-9) = 3 + 4 = 7. Correct.