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 ${C}{1}$ and ${C}{2}$ and ${r}{1}$ and ${r}{2}$ are respectively the centroids and radii of incircles of two congruent triangles, then which one of the following is correct?
  1. ${C}_{1}$ and ${C}_{2}$ are the same point and ${r}_{1}={r}_{2}$
  2. ${C}_{1}$ and ${C}_{2}$ are not necessarily the same point and ${r}_{1}={r}_{2}$.
  3. ${C}_{1}$ and ${C}_{2}$ are the same point and ${r}_{1}$ is not necessarily equal to ${r}_{2}$
  4. ${C}_{1}$ and ${C}_{2}$ are not necessarily the same point and ${r}_{1}$ is not necessarily equal to ${r}_{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

Two circles of radii $r_1$ and $r_2(r_1 > r_2)$ touch each other externally. Then the radius of circle which touches both of them externally and also their direct common tangent is?
  1. $\dfrac{r_1r_2}{\left(\sqrt{r_1}+\sqrt{r_2}\right)^2}$
  2. $\sqrt{r_1r_2}$
  3. $\dfrac{r_1+r_2}{2}$
  4. $\dfrac{r_1-r_2}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For two circles of radii r1 and r2 touching externally, the radius r of a third circle touching both and their common tangent is given by 1/sqrt(r) = 1/sqrt(r1) + 1/sqrt(r2). Solving for r gives r = r1*r2 / (sqrt(r1) + sqrt(r2))^2.

Multiple choice

The centre of the circle passing through $(6,0)$ and tangent to the circle $x^{2}+y^{2}=4$ at $(2, 0)$ is
  1. $(0,-6)$
  2. $(1,-5)$
  3. $(0,-9)$

  4. none
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The circle x^2 + y^2 = 4 has center (0,0) and radius 2. A circle tangent to it at (2,0) must have its center on the x-axis. Since it also passes through (6,0), the center must be the midpoint of the chord or satisfy the distance condition. Any circle tangent at (2,0) must have a center at (c, 0). The distance from (c,0) to (2,0) is |2-c| = radius. The distance from (c,0) to (6,0) is |6-c| = radius. Thus |2-c| = |6-c|, which has no solution (2-c = 6-c implies 2=6).

Multiple choice

lf $(0,\displaystyle \frac{5}{2})$ is a centre of similitude for the circles $\mathrm{x}^{2}+\mathrm{y}^{2}+6\mathrm{x}-2\mathrm{y}+1=0$ and $\mathrm{x}^{2}+\mathrm{y}^{2}-2\mathrm{x}-6\mathrm{y}+ 9=0$ then the length of the common tangent of the circles through it is
  1. $6$
  2. $3$
  3. $2$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The center of similitude is given as (0, 2.5). The circles have centers C1(-3, 1) and C2(1, 3) with radii r1=3 and r2=2. The length of the common tangent through the center of similitude can be calculated using the power of the point formula, which is sqrt(S1) or sqrt(S2).

Multiple choice

Two circles with radii a and b touch each other externally such that $\theta$ is the angle between the direct common tangents $(a > b \geq 2)$, then
  1. $\theta = 2 \cos^{-1} \displaystyle \left ( \frac{a-b}{a+b}\right )$
  2. $\theta = 2\tan^{-1} \displaystyle \left ( \frac{a+b}{a-b}\right )$
  3. $\theta = 2\sin^{-1} \displaystyle \left ( \frac{a+b}{a-b}\right )$
  4. $\theta = 2 \sin^{-1} \displaystyle \left ( \frac{a-b}{a+b}\right )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two circles of radii a and b touching externally, the angle theta between the direct common tangents is given by the formula sin(theta/2) = (a-b)/(a+b). Thus, theta = 2 * arcsin((a-b)/(a+b)).

Multiple choice

$C_{1}$ and $C_{2}$ are circles of unit radius with centres at $(0, 0)$ and $(1, 0)$ respectively. $C_{3}$ is a circle of unit radius, passes through the centres of the circles $C_{1}$ and $C_{2}$ and have its centre above x-axis. Equation of the common tangent to $C_{1}$ and $C_{3}$ which does not pass through $C_{2}$ is
  1. $ x -\sqrt{3}y + 2 = 0$
  2. $ \sqrt{3} x -y + 2 = 0$
  3. $\sqrt {3}x -y -2 = 0$
  4. $ x +\sqrt {3} y + 2 = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

C1 is centered at (0,0) with radius 1. C3 has center (0.5, sqrt(0.75)) = (0.5, sqrt(3)/2) and radius 1. The common tangent not passing through C2 is the line tangent to both C1 and C3, which is sqrt(3)x - y + 2 = 0.

Multiple choice

Consider a family of circles passing through two fixed points $A(3, 7)$ and $B(6, 5)$. The chord, in which the circle $x^2 + y^2 - 4x - 6y- 3 = 0$ cuts each member of the family of circles, passes through a fixed point $(a, b)$. Find the numerical value of $a + 3b$.

  1. 3

  2. 13

  3. 25

  4. 35
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The family of circles passing through A and B is S + kL = 0. The common chord of any member and the fixed circle is the radical axis. The radical axis of the family and the fixed circle passes through the intersection of the fixed circle and the line AB. Solving for the fixed point yields the result.