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 equation of a given circle is ${ x }^{ 2 }+{ y }^{ 2 }=36$, then the length of the chord which lies along the line $3x+4y-15=0$
  1. $3\sqrt { 6 } $
  2. $2\sqrt { 3 } $
  3. $6\sqrt { 3 } $

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The distance d from the center (0,0) to the line 3x+4y-15=0 is |3(0)+4(0)-15| / sqrt(3^2+4^2) = 15/5 = 3. The radius r is 6. The chord length is 2 * sqrt(r^2 - d^2) = 2 * sqrt(36 - 9) = 2 * sqrt(27) = 2 * 3 * sqrt(3) = 6 * sqrt(3).

Multiple choice

Two chords PQ and MN of length 11 cm and 5 cm respectively of a circle are parallel to each other and are one the opposite sides of its centre. If the distance between chord MN and chord PQ is 6 cm find the radius of the circle
  1. $\displaystyle \frac{10 \sqrt{5}}{2} cm$
  2. $\displaystyle \frac{5 \sqrt{5}}{4} cm$
  3. $\displaystyle \frac{5 \sqrt{5}}{2} cm$
  4. $\displaystyle \frac{5 \sqrt{10}}{2} cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let r be the radius. Distance from center to chord PQ (length 11) is sqrt(r^2 - 5.5^2). Distance to chord MN (length 5) is sqrt(r^2 - 2.5^2). Since they are on opposite sides, the sum of distances is 6. Solving sqrt(r^2 - 30.25) + sqrt(r^2 - 6.25) = 6 leads to r = 5*sqrt(5)/2.

Multiple choice

The equation to the circle which is touched by $y=x$, has its centre on the positive direction of the x axis and cuts off a chord of length $2$ units along the line $\displaystyle y-\frac { x }{ \sqrt { 3 } } =0$
  1. ${ x }^{ 2 }+{ y }^{ 2 }-4x+2=0$
  2. ${ x }^{ 2 }+{ y }^{ 2 }-4x+1=0$
  3. ${ x }^{ 2 }+{ y }^{ 2 }-8x+8=0$
  4. ${ x }^{ 2 }+{ y }^{ 2 }-4y+2=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Circle center (h, 0), radius r. Touches y=x (x-y=0), so distance from (h,0) to x-y=0 is r: |h-0|/sqrt(2) = r, so h^2 = 2r^2. Chord length 2 on y - x/sqrt(3) = 0. Distance from center to line is sqrt(r^2 - 1^2). Solving leads to x^2 + y^2 - 4x + 2 = 0.

Multiple choice

A rhombus is formed by two radii and two chords of a circle whose radius is $16$cm. The area of the rhombus in sq.cm is.
  1. $128$
  2. $128\sqrt3$
  3. $256$
  4. $512$
  5. $512\sqrt3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A rhombus formed by two radii and two chords in a circle of radius R must have sides of length R. Since the radii form an angle at the center, the rhombus is composed of two equilateral triangles with side 16. Area = 2 * (sqrt(3)/4 * 16^2) = 128 * sqrt(3).

Multiple choice

The circles ${ S }{ 1 }$ with centre ${ C }{ 1 }\left( { a }{ 1 },{ b }{ 1 } \right) $ and radius ${ r }{ 1 }$ touches externally the circle ${ S }{ 2 }$ with centre ${ C }{ 2 }\left( { a }{ 2 },{ b }{ 2 } \right) $ and radius ${ r }{ 2 }$. If the tangent at their common point passes through the origin, then
  1. $\left( { a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 } \right) +\left( { b }_{ 1 }^{ 2 }+{ b }_{ 2 }^{ 2 } \right) ={ r }_{ 1 }^{ 2 }+{ r }_{ 2 }^{ 2 }$
  2. $\left( { a }_{ 1 }^{ 2 }-{ a }_{ 2 }^{ 2 } \right) +\left( { b }_{ 1 }^{ 2 }-{ b }_{ 2 }^{ 2 } \right) ={ r }_{ 1 }^{ 2 }-{ r }_{ 2 }^{ 2 }$
  3. ${ \left( { a }_{ 1 }^{ 2 }-{ b }_{ 2 } \right) }^{ 2 }+\left( { a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 } \right) ={ r }_{ 1 }^{ 2 }+{ r }_{ 2 }^{ 2 }$
  4. $\left( { a }_{ 1 }^{ 2 }-{ b }_{ 1 }^{ 2 } \right) +\left( { a }_{ 1 }^{ 2 }+{ b }_{ 2 }^{ 2 } \right) ={ r }_{ 1 }^{ 2 }+{ r }_{ 2 }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two circles touching externally, the distance between centers equals the sum of radii: (a1-a2)^2 + (b1-b2)^2 = (r1+r2)^2. The tangent at the common point passes through the origin. This geometric condition leads to the relation (a1^2 - a2^2) + (b1^2 - b2^2) = r1^2 - r2^2.

Multiple choice

If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2 r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement true or false? Why?
  1. $ false,\quad since\quad the\quad radius\quad of\quad { C }_{ 1 }\neq the\quad radius\quad of{ \quad C }_{ 2 }.\quad \\ $
  2. $ True,\quad since\quad the\quad radius\quad of\quad { C }_{ 1 }=2\times (the\quad radius\quad of{ \quad C }_{ 2 }).\quad \quad \\ $
  3. $ False,\quad \quad since\quad \theta _{ 1 }\neq \theta _{ 2 }.\\ $
  4. $ True,\quad since\quad \theta _{ 1 }=\theta _{ 2 }.\\ $
Reveal answer Fill a bubble to check yourself
B Correct answer
Multiple choice

Let $C$ be a circle with center '$ O$' and $HK$ is the chord of contact of pair of the tangents from point $A$. $OA$ intersects the circles $C$ at $P$ and $Q$ and $B$ is the midpoint of $HK$, then Statement - 1: $AB$ is the harmonic mean $AP$ and $AQ$ Because Statement - 2: $AK$ is the Geometric mean of $AB$ and $AQ$ and $OA$ is the arithmetic mean of $AP$ and $AQ$.

  1. Statement - 1 is true, statement -2 is true and statement -2 is correct explanation for statement -1.

  2. Statement - 1 is true, statement -2 is true and statement -2 is NOT the correct explanation for statement -1.

  3. Statement - 1 is true, statement -2 is false

  4. Statement - 1 is false , statement -2 is true
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

STATEMENT - 1 : Locus of mid point of chords of circle $x^{2} + y^{2} = 4$ which subtends angle of $\displaystyle \frac{\pi }{2}$ at origin is $x^{2} + y^{2} = 1.$ STATEMENT - 2 : If any chord of circle $x^{2} + y^{2} = r^{2}$ subtends an angle $'\theta '$ at center, then its mid point always lies on $x^{2} + y^{2} = r^{2} \cos^{2} \left ( \displaystyle \frac{\theta }{2} \right )$

  1. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

  2. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

  3. STATEMENT-1 is True, STATEMENT-2 is False

  4. STATEMENT-1 is False, STATEMENT-2 is True
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Statement 2 is a known theorem: the locus of the midpoint of a chord subtending angle theta at the center is a circle of radius r*cos(theta/2). For theta=pi/2, radius = r*cos(pi/4) = r/sqrt(2). Here r=2, so radius = 2/sqrt(2) = sqrt(2). The locus is x^2 + y^2 = 2. Statement 1 claims x^2 + y^2 = 1, which is false.

Multiple choice

Equation to the circle whose one of the diameters is the common chord of $(\mathrm{x}-\mathrm{a})^{2}+\mathrm{y}^{2}=\mathrm{a}^{2},\ \mathrm{x}^{2}+(\mathrm{y}-\mathrm{b})^{2}=\mathrm{b}^{2}$ is
  1. $(\mathrm{a}^{2}+\mathrm{b}^{2})(\mathrm{x}^{2}+\mathrm{y}^{2})=2\mathrm{a}\mathrm{b}(\mathrm{b}\mathrm{x}+\mathrm{a}\mathrm{y})$
  2. $(\mathrm{a}^{2}+\mathrm{b}^{2})(\mathrm{x}^{2}+\mathrm{y}^{2})=2\mathrm{a}\mathrm{b}(\mathrm{a}\mathrm{x}+\mathrm{b}\mathrm{y})$
  3. $\mathrm{x}^{2}+\mathrm{y}^{2}=\dfrac {2\mathrm{a}\mathrm{b}}{(\mathrm{a}^{2}+\mathrm{b}^{2}) (ax-by)}$
  4. $\mathrm{x}^{2}+\mathrm{y}^{2}= \dfrac {ab}{(\mathrm{a}^{2}+\mathrm{b}^{2})(\mathrm{a}\mathrm{x}+\mathrm{b}\mathrm{y})}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The common chord of the two circles is found by subtracting the equations, resulting in ax + by = 0. The equation of a circle passing through the intersection of two circles is S1 + k(S2-S1) = 0, or more simply, the family of circles passing through the intersection. Using the condition that the common chord is a diameter, we derive the correct equation.

Multiple choice

Fill in the blanks: (i) An arc is the part of the ......., (ii) Diameter of a circle bisects ......, (iii) The part of the circumference greater than the semi-circle is called........ (iv) Sector of a circle is its region bounded by ........., (v) The segment of a circle is the region bounded by ........, (vi) A tangent of a circle meets the circle at ......., (vii) The number of tangents that can be drawn through a point on its circumference ........, (viii) The number of tangents that can be drawn through a point outside the circle is ......

  1. (i) circumference. (ii) the circle. (iii) major arc. (iv) two radii and the corresponding arc. (v) a chord and the corresponding arc. (vi) one and only one point. (vii) is one and only one. (viii) two.

  2. (i) circumference.(ii) the circle.(iii)  major segment.(iv) two radii and the corresponding arc.(v) a chord and the corresponding arc. (vi) one and only one point.(vii) is two and only two.(viii) one. 

  3.  (i) circumference. (ii) the diameter. (iii) major arc.(iv) two radii and the corresponding semicircle.(v) a diameter and the corresponding arc.(vi) two and only two point.(vii) is two and only two.(viii)\quad two. 

  4.  (i) circumference. (ii) the circle.(iii) major arc.(iv) two radii and the corresponding chord.(v) a chord and the corresponding diameter.(vi) two and only two point.(vii) is two and only  two.(viii) two. 
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The definitions provided in option A are geometrically accurate: an arc is part of a circumference, diameter bisects a circle, sector is bounded by two radii and an arc, and tangents from an external point are two.

Multiple choice

With reference to a given circle, $\displaystyle A_{1}$ and $\displaystyle B_{1}$ are the areas of the inscribed and circumscribed regular polygons of n sides, $\displaystyle A_{2}$ and $\displaystyle B_{2}$ are corresponding quantities for regular polygons of 2n sides (a)If $\displaystyle A_{2}$ is a geometric mean between $\displaystyle A_{1}$ and $\displaystyle B_{1}$ then value of $A_1B_1$ (b) If$\displaystyle B_{2}$ is a harmonic mean between $\displaystyle A_{2}$ and $\displaystyle B_{1}$ then value of $\cfrac{1}{A_1}+\cfrac{1}{B_1}$
  1. $a:n^{2}R^{2}\sin^{2}\cfrac{\pi}{n},b:\cfrac{2}{B_{1}}$
  2. $a:n^{2}R^{4}\sin^{2}\cfrac{\pi}{n},b:\cfrac{2}{B_{2}}$
  3. $a:n^{2}R^{2}\sin^{2}\cfrac{\pi}{n},b:\cfrac{1}{B_{1}}$
  4. $a:n^{2}R^{4}\sin^{2}\cfrac{\pi}{n},b:\cfrac{1}{B_{2}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is a standard geometric property of inscribed and circumscribed polygons. For part (a), A1B1 = n^2 * R^4 * sin^2(pi/n). For part (b), the harmonic mean relation leads to 2/B2 = 1/A2 + 1/B1, which rearranges to the required form.

Multiple choice

Consider ${L}{1}:2{x}+3{y}+{p}-3=0$ ${L}{2}:2{x}+3{y}+{p}+3=0$, where ${p}$ is a real number, and ${C}:{x}^{2}+{y}^{2}+6{x}-10{y}+30=0$. STATEMENT 1 : If line $L_{1}$ is a chord of circle $C$, then line $L_{2}$ is not always a diameter of circle $C$. STATEMENT 2 : If line $L_{1}$ is a diameter of circle $C$, then line $L_{2}$ is not a chord of circle $C$.

  1. STATEMENT 1 is True, STATEMENT 2 is True; STATEMENT 2 is a correct explanation for STATEMENT 1.

  2. STATEMENT 1 is True, STATEMENT 2 is True; STATEMENT 2 is NOT a correct explanation for STATEMENT 1.

  3. STATEMENT 1 is True, STATEMENT 2 is False.

  4. STATEMENT 1 is False, STATEMENT 2 is True.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle C has center (-3, 5) and radius sqrt(9 + 25 - 30) = 2. A line 2x + 3y + k = 0 is a diameter if it passes through (-3, 5), meaning 2(-3) + 3(5) + k = 0, so k = -9. For L1, p-3 = -9 implies p = -6. For L2, p+3 = -9 implies p = -12. Since p can vary, L1 being a chord does not force L2 to be a diameter, making Statement 1 true. Statement 2 is false because if L1 is a diameter, L2 is just another line with a different constant term, which could also be a chord or diameter depending on p.

Multiple choice

If chord contact of the tangents drawn from the point $\left( \alpha ,\beta \right) $ to the ellipse $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1,$ touches the circle ${ x }^{ 2 }+{ y }^{ 2 }={ c }^{ 2 },$ THEN THE LOCUS OF THE POINT $\left( \alpha ,\beta \right) $
  1. $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =\dfrac { 1 }{ { c }^{ 2 } } $
  2. $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =\dfrac { 1 }{ { c }^{ 4 } } $
  3. $\dfrac { { x }^{ 2 } }{ { a }^{ 4 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 4 } } =\dfrac { 1 }{ { c }^{ 2 } } $

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The chord of contact of tangents from (alpha, beta) to the ellipse x^2/a^2 + y^2/b^2 = 1 is (alpha*x)/a^2 + (beta*y)/b^2 = 1. If this line touches the circle x^2 + y^2 = c^2, the perpendicular distance from the origin to the line must equal the radius c. Thus, 1 / sqrt(alpha^2/a^4 + beta^2/b^4) = c, leading to alpha^2/a^4 + beta^2/b^4 = 1/c^2.

Multiple choice

Consider a circle with unit radius whose chord AB is at a perpendicular distance x from centre C. Let area of triangle ACB is given by function y = f(x). On the curve y = f(x) a variable point P(x,f(x)) is taken and perpendiculars PQ and PR are drawn on x & y axis respectively. Area of rectangle QPRO is given by y = g(x). (where 'O' is origin). List I List II P. The greatest area of $\triangle$ ACB is 1. $\displaystyle \frac{2}{3 \sqrt{3}}$ Q. The greatest area of rectangle QPRO is 2. $\displaystyle \frac{\sqrt{13} -1 }{6}$ R. The greatest vertical distance between y = (x) & y = g(x) occurs at x equal to 3. $\displaystyle \frac{1}{3}$ S. The area bounded by y = f(x), y = 0 & x = 1 is 4. $\displaystyle \frac{1}{2}$

  1. P -1, Q -2, R - 3, S - 4

  2. P -4, Q - 3, R - 2, S - 1

  3. P -3, Q - 1, R - 2, S - 4

  4. P - 4, Q - 1, R - 2, S - 3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

This is a standard calculus/geometry problem involving optimization of areas. P corresponds to the maximum area of the triangle (4), Q to the rectangle (1), R to the distance (2), and S to the bounded area (3).