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

Locus of the mid point of the chord of circle $\displaystyle x^{2}+y^{2}=16 $ which is subtending right angle at the point (5, 0) is
  1. $\displaystyle x^{2}+y^{2}+4x+5/2=0 $
  2. $\displaystyle x^{2}+y^{2}-5x+9/2=0 $
  3. $\displaystyle x^{2}+y^{2}+5x+7=0 $
  4. $\displaystyle x^{2}+y^{2}+2x+5=0 $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let (h, k) be the midpoint of the chord. The equation of the chord with midpoint (h, k) is hx + ky = h^2 + k^2. The chord subtends a right angle at (5, 0). This is a standard locus problem for a circle chord. The resulting equation is x^2 + y^2 - 5x + 9/2 = 0.

Multiple choice

Which of the following pairs of lines can be parallel? 1.Two tangents to a circle 2.Two diameters of a circle 3.A chord of circle and a tangents to a circle 4.Two chords of a circle Select the correct answer using the codes given below: Codes:

  1. 1,2 and 3

  2. 2,3 and 4

  3. 1,3 and 4

  4. 1,2 and 4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Two tangents to a circle can be parallel if they are at opposite ends of a diameter. Two chords can be parallel (e.g., chords equidistant from the center). A chord and a tangent can be parallel if the chord is parallel to the tangent at the point of contact. Two diameters must intersect at the center, so they cannot be parallel.

Multiple choice

Choose the correct answer from the alternatives given. In a circle of radius $17$ cm two parallel chords are present on the opposite side of the diameter. If the distance between them is $23$ cm and the length of one chord is $16$ cm then the length of other chord is:
  1. $15$ cm
  2. $20$ cm
  3. $18$ cm
  4. $30$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the chords be AB and CD. Distance from center to chord AB (length 16) is sqrt(17^2 - 8^2) = sqrt(289 - 64) = 15. Since the chords are on opposite sides, the distance to the other chord is 23 - 15 = 8. The half-length of the other chord is sqrt(17^2 - 8^2) = sqrt(289 - 64) = 15. The full length is 15 * 2 = 30 cm.

Multiple choice

Given two circles intersecting orthogonally having length of common chord $\dfrac {24}5$ units. Radius of one ofthe circle is $3$ units. Radius of the circle is

  1. 6 units

  2. 5 units

  3. 2 units

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

For two circles of radii r1 and r2 intersecting orthogonally, the distance between centers d satisfies d^2 = r1^2 + r2^2. The common chord length is 2*h where h is the altitude to the line of centers. Here, h = (24/5)/2 = 2.4. Using the triangle formed by radius, distance to chord, and half-chord, r1^2 = d1^2 + 2.4^2. Solving the geometry leads to r2 = 4.

Multiple choice

If $x=3$ is the chord of contact of the circle $x^{2}+y^{2}=81$, then the equation of the corresponding pair of tangents, is
  1. $x^{2}-8y^{2}+54x+729=0$
  2. $x^{2}-8y^{2}-54x+729=0$
  3. $x^{2}-8y^{2}-54x-729=0$
  4. $x^{2}-8y^{2}=729$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a circle x^2 + y^2 = r^2, the chord of contact for a point (x1, y1) is xx1 + yy1 = r^2. Here, the chord is x = 3, so x(3) = 81 implies x = 27. The point is (27, 0). The pair of tangents is given by SS1 = T^2, where S = x^2 + y^2 - 81 and T = 27x - 81. Substituting and simplifying yields x^2 - 8y^2 - 54x + 729 = 0.

Multiple choice

If a circle of radius $3$ units touching the lines $\sqrt { 3 } { y }^{ 2 }-4xy+\sqrt { 3 } { x }^{ 2 }=0$ in the first quadrant then the length of the chord of contact to this circle
  1. $\dfrac { \sqrt { 3 } +1 }{ 2 }$
  2. $\dfrac { \sqrt { 3 } +1 }{ \sqrt { 2 } }$
  3. $3\left( \dfrac { \sqrt { 3 } +1 }{ \sqrt { 2 } } \right)$
  4. $\dfrac { \left( \sqrt { 3 } +1 \right) }{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The given lines represent a pair of straight lines. By simplifying the equation, we find the angle between them and the distance from the origin. The circle touches these lines, and the chord of contact is calculated using the geometry of the circle and the lines.

Multiple choice

From an external point $P,\ a$ tangent $PT$ and a line segment $PAB$ is drawn to circle with centre $O,\ ON$ is perpendicular to the chord $AB$. then $PA.PB=PN^{2}-AN^{2}$. then statement is ?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By the Power of a Point theorem, PT^2 = PA * PB. In the right triangle ONA, OA^2 = ON^2 + AN^2. Since OA is the radius R, R^2 = ON^2 + AN^2. The geometric relationship PA * PB = PT^2 is standard, and the expression provided relates to the geometry of chords and tangents.

Multiple choice

Middle point of the chord of the circle $x^2+y^2=25$ intercepted on the line $x-2y=2$ is
  1. $\left ( \dfrac{3}{5},\dfrac{4}{5} \right )$
  2. $(-2,-2)$
  3. $\left ( \dfrac{2}{5},-\dfrac{4}{5} \right )$
  4. $\left ( \dfrac{8}{3},\dfrac{1}{3} \right )$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The midpoint (h, k) of a chord of x^2 + y^2 = r^2 with equation lx + my = n is given by the condition that the line joining the origin to the midpoint is perpendicular to the chord. The line x - 2y = 2 has slope 1/2. The perpendicular line through the origin has slope -2, so y = -2x. Substituting into x - 2y = 2: x - 2(-2x) = 2, 5x = 2, x = 2/5. Then y = -4/5. Midpoint is (2/5, -4/5).