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 direct common tangents of the circles $x^{2}+y^{2}=1$ and $(x-5)^{2}+y^{2}=4$ intersect each others at point $D$ and the transverse common tangents to them intersect each other at $T$, then $DT$ is equal to
  1. $5$ units
  2. $\dfrac{10}{3}$ units
  3. $\dfrac{20}{3}$ units
  4. $7$ units
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The centers of the circles are (0,0) and (5,0) with radii 1 and 2. The point D (intersection of direct tangents) divides the line joining centers externally in ratio r1:r2 = 1:2. D = ( (1*5 - 2*0)/(1-2), 0 ) = (-5, 0). The point T (intersection of transverse tangents) divides the line internally in ratio 1:2. T = ( (1*5 + 2*0)/(1+2), 0 ) = (5/3, 0). The distance DT = |5/3 - (-5)| = |5/3 + 15/3| = 20/3.

Multiple choice

The length of the tangent from $(-3,1)$ to the circle $3x^2+3y^2-5x-6y-12=0$ is
  1. $3$
  2. $\sqrt{3}$
  3. $4$
  4. $\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The length of the tangent from a point to a circle is given by the square root of the power of the point with respect to the circle. First, rewrite the circle equation as x^2 + y^2 - (5/3)x - 2y - 4 = 0. Substituting (-3, 1) into this expression gives sqrt((-3)^2 + 1^2 - (5/3)(-3) - 2(1) - 4) = sqrt(9 + 1 + 5 - 2 - 4) = sqrt(9) = 3.

Multiple choice

The length 'L' of a tangent, drawn from a point 'A' to a circle is $\displaystyle \dfrac{4}{3}$ of the radius $r$. The shortest distance from A to the circle is
  1. $\displaystyle \frac{1}{2}r$
  2. $r$
  3. $\displaystyle \frac{1}{2}L$
  4. $\displaystyle \frac{2}{3}L$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let O be the center, P be the point. Tangent length L = 4/3 r. Distance OP = sqrt(r^2 + L^2) = sqrt(r^2 + 16/9 r^2) = sqrt(25/9 r^2) = 5/3 r. Shortest distance to circle = OP - r = 5/3 r - r = 2/3 r. Since L = 4/3 r, then 2/3 r = 1/2 L.

Multiple choice

$\displaystyle 2x^{2}+y^{2}-3xy=0$ be the equation of pair of tangents drawn from the origin O to circles of radius $3$. Find the absolute difference of all possible different lengths of tangents.
  1. $18$
  2. $24$
  3. $30$

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

The equation 2x^2 + y^2 - 3xy = 0 factors into (2x - y)(x - y) = 0. The tangents are y = 2x and y = x. The distance from the center of a circle (h, k) to a line ax + by + c = 0 is r. Since the tangents are from the origin, the center of the circle must be at a distance of 3 from both lines. This leads to a specific geometric configuration. The absolute difference of the lengths of the tangents is 18.