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 length of the common chord of the circles $x^2 + y^2 = 12$ and $x^2 + y^2 - 4x + 3y - 2 = 0$, is
  1. $4\sqrt{2}$
  2. $5 \sqrt{2}$
  3. $2 \sqrt{2}$
  4. $6 \sqrt{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The common chord equation is S1 - S2 = 0. (x^2 + y^2 - 12) - (x^2 + y^2 - 4x + 3y - 2) = 0, which is 4x - 3y - 10 = 0. Distance from center (0,0) to chord is |4(0) - 3(0) - 10| / sqrt(4^2 + 3^2) = 10/5 = 2. Chord length = 2 * sqrt(r^2 - d^2) = 2 * sqrt(12 - 4) = 2 * sqrt(8) = 4 * sqrt(2).

Multiple choice

A circle of radius r is concentric with $\dfrac{x^2}{a^2}$ + $\dfrac{y^2}{b^2}$ = 1. The slope of a common tangent to them is
  1. $\sqrt{r^2 b^2}$
  2. $\sqrt{r^2 a^2}$
  3. $\sqrt{r^2 + b^2}$
  4. $\sqrt{\dfrac{b^2-  r^2}{r^2 -a^2}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The condition for a line y = mx + c to be tangent to the ellipse x^2/a^2 + y^2/b^2 = 1 is c^2 = a^2m^2 + b^2. For a circle x^2 + y^2 = r^2, the condition is c^2 = r^2(1 + m^2). Equating the two: a^2m^2 + b^2 = r^2 + r^2m^2. Solving for m^2: m^2(a^2 - r^2) = r^2 - b^2, so m = sqrt((b^2 - r^2) / (r^2 - a^2)).

Multiple choice

The length of the chord of the circle $x^{2}+y^{2}=9$ passing through $(3, 0)$ and perpendicular to the line $y+x=0$ is
  1. $\dfrac {3}{\sqrt{2}}$
  2. $3\sqrt{2}$
  3. $2\sqrt{3}$

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

The line perpendicular to y + x = 0 (slope -1) passing through (3, 0) is y - 0 = 1(x - 3), or y = x - 3. The distance from the center (0, 0) to this line x - y - 3 = 0 is d = |0 - 0 - 3| / sqrt(1^2 + (-1)^2) = 3 / sqrt(2). The chord length is 2 * sqrt(r^2 - d^2) = 2 * sqrt(9 - 9/2) = 2 * sqrt(9/2) = 2 * 3 / sqrt(2) = 3 * sqrt(2).

Multiple choice

The equation of diameter of the circle $\displaystyle (x - 2)^2 + (y + 1)^2 = 16$ which bisect the chord cut off by the circle on the line $\displaystyle x - 2y - 3 = 0$ is
  1. $\displaystyle 2x - y + 3 = 0$
  2. $\displaystyle 2x + y - 3 = 0$
  3. $\displaystyle 2x + y + 3 = 0$

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

The center of the circle is (2, -1). The diameter must pass through this center. The midpoint of the chord on the line x - 2y - 3 = 0 is found by dropping a perpendicular from the center (2, -1) to the line. The line perpendicular to x - 2y - 3 = 0 is 2x + y + k = 0. Since it passes through (2, -1), 2(2) + (-1) + k = 0, so k = -3. The diameter is 2x + y - 3 = 0.

Multiple choice

Let $x(x-a)+y(y-1)$ be a circle. If two chords from $(a,1)$ bisected by x-axis are drawn to the circle then the condition is ____________.
  1. ${ a }^{ 2 }>8$
  2. ${ a }^{ 2 }<8$
  3. ${ a }^{ 2 }>4$
  4. ${ a }^{ 2 }<4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circle is x^2 - ax + y^2 - y = 0. Center is (a/2, 1/2). A chord from (a, 1) bisected by the x-axis (y=0) means the midpoint is (h, 0). The chord is a line through (a, 1) and (h, 0). The condition for such chords to exist involves the distance from the center to the chord being less than the radius.

Multiple choice

If there are two chords of the circle $ x ^ { 2 } + y ^ { 2 } - 2 x - 4 y - 20 = 0 $ which passes through ( h , k ) and bisected by x-axis then ____________.
  1. $

    ( h - 1 ) ^ { 2 } > 8 k

    $
  2. $

    \mathrm { h } ^ { 2 } > 8 \mathrm { k }

    $
  3. $

    ( h + 1 ) ^ { 2 } > 8 k

    $
  4. $

    k ^ { 2 } > 8 h

    $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A chord passing through (h, k) and bisected by the x-axis means the midpoint is (x, 0). The chord equation is derived from the circle equation. The condition for the chord to exist and be bisected leads to the inequality (h-1)^2 > 8k.

Multiple choice

PQ and RS are two parallel chords of a circle whose centre is O and radius is $10$ cm. If PQ $= 16$ cm and RS $= 12$ cm. Then the distance between PQ and RS, if they lie. (i) on the same side of the centre O and (ii) on the opposite of the centre O are respectively
  1. $8$ cm & $14$ cm
  2. $4$ cm & $14$ cm
  3. $2$ cm & $14$ cm
  4. $2$ cm & $28$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Distance of chord from center d = sqrt(r^2 - (chord/2)^2). For PQ: d1 = sqrt(100 - 64) = 6. For RS: d2 = sqrt(100 - 36) = 8. Same side: |8 - 6| = 2. Opposite side: 8 + 6 = 14.