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

How many tangents can a circle have?
  1. $1$
  2. $2$
  3. $8$

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

A circle is a collection of infinite points, and at each point on the circumference, a unique tangent can be drawn.

Multiple choice

If $O=(0,0),A=(1,0)$ and $B=\left(\dfrac {1}{2},\dfrac {\sqrt {3}}{2}\right)$ then centre of the circle for which the lines $OA,OB$ and $AB$ are tangents is
  1. $\left(\dfrac {1}{2},\dfrac {1}{2\sqrt {3}}\right)$
  2. $\left(\dfrac {1}{2},\dfrac {1}{1\sqrt {3}}\right)$
  3. $\left(\dfrac {1}{\sqrt {3}},\dfrac {1}{\sqrt {2}}\right)$
  4. $\left(\dfrac {1}{2\sqrt {3}},\dfrac {1}{\sqrt {2}}\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The lines are OA (y=0), OB (y = sqrt(3)x), and AB (y - 0 = (sqrt(3)/2 - 0)/(1/2 - 1) * (x - 1), which is y = -sqrt(3)(x-1)). The center of the incircle of triangle OAB is needed. The vertices are (0,0), (1,0), (1/2, sqrt(3)/2). This is an equilateral triangle with side length 1. The incenter is (1/2, (sqrt(3)/2)/3) = (1/2, 1/(2*sqrt(3))).

Multiple choice

Consider lines $ax+2by+b=2a+3by$ a, b$\in$R, then?
  1. A unique line of above family is tangent to circle $x^2+y^2+x-y-6=0$
  2. Two lines from above family are tangent to circle $x^2+y^2-6x-4y-1=0$
  3. If above lines are pair of tangents to $x^2+y^2-x-y-1=0$, then length of tangent is $1$ unit
  4. Circumcircle of triangle formed by above lines are pair of tangents of $x^2+y^2+4x+2y+1=0$ and corresponding chord of contact is $x^2+y^2=5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

One of the sides of a triangle is divided into segments of 6 and 8 units by the point of tangency of the inscribed circle. If the radius of the circle is 4, then the length of the shortest side of the triangle is :

  1. 12 units

  2. 13 units

  3. 14 units

  4. 15 units

  5. 16 units
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the segments be x=6, y=8, z. The sides are a=x+y=14, b=y+z, c=z+x. Semi-perimeter s = (14 + y+z + z+x)/2 = (14 + 8+z + z+6)/2 = 14+z. Area = sqrt(s(s-a)(s-b)(s-c)) = sqrt((14+z)(z)(6)(8)) = sqrt(48z(14+z)). Also Area = rs = 4(14+z). Squaring: 48z(14+z) = 16(14+z)^2. 3z = 14+z, 2z = 14, z = 7. Sides are 14, 15, 13. Shortest is 13.

Multiple choice

Obtain the equations of the straight line passing through the point $A(2,0)$ & making ${45}^{o}$ angle with the tangent at $A$ to the circle ${(x+2)}^{2}+{(y-3)}^{3}=25$. Find the equations of the circles each of radius $3$ whose centres are on these straight lines at a distance of $5\sqrt {2}$ from $A$.
  1. $x-7y=2, 7x+y=14;$
  2. ${(x-1)}^{2}+{(y-7)}^{2}={3}^{2};$ ${(x-3)}^{2}+{(y+7)}^{2}={3}^{2}$
  3.  ${(x-9)}^{2}+{(y-1)}^{2}={3}^{2};$ ${(x+5)}^{2}+{(y+1)}^{2}={3}^{2}$
  4.  ${(x-9)}^{2}+{(y-1)}^{2}={3}^{2};$ ${(x-5)}^{2}+{(y-1)}^{2}={3}^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Multiple choice

The tangents at the extremities of each of two chords drawn from the point $(2, 2)$ to the circle $\displaystyle x^{2}+y^{2}-2x-4y+1=0 $ meet at points A and B, respectively. The equation of the line AB is
  1. $x = 5$
  2. $x + y = 4$
  3. $2x - y = 2$
  4. $y = 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord of contact from a point (x1, y1) to a circle x^2+y^2+2gx+2fy+c=0 is xx1 + yy1 + g(x+x1) + f(y+y1) + c = 0. For (2,2) and the circle, the chord equation is 2x + 2y - 1(x+2) - 2(y+2) + 1 = 0, which simplifies to x - 5 = 0, or x = 5.

Multiple choice

If the equation of pair of direct common tangent to the circles ${ \left( x+2 \right) }^{ 2 }+{ y }^{ 2 }=1$ and ${ \left( x-4 \right) }^{ 2 }+{ y }^{ 2 }=9$ is given by ${ x }^{ 2 }+2hxy+b{ y }^{ 2 }+2gx+2fy+c=0$. Then ${ b }^{ 2 }-c$ is equal to
  1. $39$
  2. $93$
  3. $36$
  4. $35$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The direct common tangents to circles (x+2)^2 + y^2 = 1 and (x-4)^2 + y^2 = 9. The centers are (-2, 0) and (4, 0) with radii 1 and 3. The external center of similitude divides the centers in ratio 1:3 externally: ((1*4 - 3*-2)/(1-3), 0) = (10/-2, 0) = (-5, 0). The tangent lines pass through (-5, 0). Using the condition for tangency, the equation leads to b^2 - c = 39.

Multiple choice

From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. What are the lengths (in cm) of tangents PQ and PR ?

  1. 13, 12

  2. 13, 13

  3. 12, 12

  4. 12, 18
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The radius, the tangent, and the line from the center to the external point form a right-angled triangle. Hypotenuse = 13, one leg = 5. Tangent length = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12.