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

The mid point of the chord $x-2y+7=0$ w.r.t the circle $x^2+y^2-2x-10y+1=0$ is?
  1. $(7, 21)$
  2. $\left(\dfrac{7}{4}, \dfrac{21}{4}\right)$
  3. $\left(\dfrac{7}{5}, \dfrac{21}{5}\right)$
  4. $\left(\dfrac{-7}{5}, \dfrac{-21}{5}\right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The center of the circle is (1, 5). The midpoint of a chord is the projection of the center onto the line x - 2y + 7 = 0. The line passing through (1, 5) perpendicular to the chord is 2x + y - 7 = 0; solving the system of these two lines gives (7/5, 21/5).

Multiple choice

If $OA$ and $OB$ are two perpendicular chords of the circle $r=a\cos { \theta } +b\sin { \theta }$ passing through origin, then the locus of the mid point of $AB$ is:
  1. ${x}^{2}+{y}^{2}=\dfrac{ax}2+\dfrac {by}2$
  2. $x=a/2$
  3. ${x}^{2}-{y}^{2}={a}^{2}+{b}^{2}$
  4. $y=b/2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation r = a*cos(theta) + b*sin(theta) represents a circle passing through the origin. Converting to Cartesian: x^2 + y^2 = ax + by. The midpoint of perpendicular chords OA and OB through the origin satisfies the locus x^2 + y^2 = (ax/2) + (by/2).

Multiple choice

if the angle between the radii of a circle is ${130}^{0}$ then the angle between the tangents at the ends of the radii is
  1. ${90}^{0}$
  2. ${70}^{0}$
  3. ${50}^{0}$
  4. ${40}^{0}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The angle between the radii and the tangents at their endpoints is 90 degrees. In the quadrilateral formed by the center, the two points of tangency, and the intersection of the tangents, the sum of angles is 360 degrees. The angle between the tangents is 180 - 130 = 50 degrees.

Multiple choice

If t is the length of tangent to the circle $\displaystyle x^{2}+y^{2}+2g_{i}x+5=0,i=1,2,3$ from any point and $\displaystyle g_{1},g_{2},g_{3}$ are in AP and $\displaystyle A_{i}\left ( g_{i}-t_{i}^{2} \right )$ Then
  1. $\displaystyle A_{1},A_{2},A_{3}$ are collinear
  2. $\displaystyle t_{1}^{2},t_{2}^{2},t_{3}^{2}$ are in GP
  3. $\displaystyle A_{1},A_{2},A_{3}$ form equilateral triangle
  4. $\displaystyle A_{1},A_{2},A_{3}$ form right angle triangle
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

Angle between tangents drawn to circle ${ x }^{ 2 }+{ y }^{ 2 }=20$, from the point $\left( 6,2 \right) $ is
  1. $\dfrac { \pi }{ 2 } $
  2. $\pi $
  3. $\dfrac { \pi }{ 4 } $
  4. $2\pi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The point (6, 2) is outside the circle x^2 + y^2 = 20 (since 36 + 4 = 40 > 20). The distance from the center (0,0) to (6,2) is sqrt(40). The radius is sqrt(20). Let theta be the angle between the tangent and the line connecting the center to the point. sin(theta) = r/d = sqrt(20)/sqrt(40) = 1/sqrt(2). So theta = 45 degrees. The total angle between the two tangents is 2 * theta = 90 degrees, which is pi/2.

Multiple choice

Tangents are drawn to the circles $x^{2} + y^{2} + 4x + 6y - 19 = 0$ and $x^{2} + y^{2} = 9$ from any point on the line $2x + 3y = 5$.then their lengths are equal.

  1. True

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

The length of the tangent from a point (x1, y1) to a circle is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). For the first circle, L1^2 = x1^2 + y1^2 + 4x1 + 6y1 - 19. For the second, L2^2 = x1^2 + y1^2 - 9. If the point lies on 2x + 3y = 5, then 2x1 + 3y1 = 5. Substituting 4x1 + 6y1 = 2(2x1 + 3y1) = 10 into L1^2: L1^2 = x1^2 + y1^2 + 10 - 19 = x1^2 + y1^2 - 9. Since L1^2 = L2^2, the lengths are equal.