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 tangent from $(1, 1)$ to the circle $2x^2+2y^2+5x+3y+1=0$ is?
  1. $\sqrt{\dfrac{13}{2}}$
  2. $3$
  3. $2$
  4. $1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The length of the tangent from (x1, y1) to a circle x^2 + y^2 + gx + fy + c = 0 is sqrt(x1^2 + y1^2 + gx1 + fy1 + c). First, normalize the equation by dividing by 2: x^2 + y^2 + 2.5x + 1.5y + 0.5 = 0. Plugging in (1, 1) gives sqrt(1 + 1 + 2.5 + 1.5 + 0.5) = sqrt(6.5) = sqrt(13/2).

Multiple choice

The angle between the tangents to the circle with centre $(4,5) $ drawn from $P(-2,-3) $ is $120^o$ , the length of the tangent to the circle from $P$ is :
  1. $4$
  2. $3$
  3. $2$
  4. $5$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Distance from P(-2,-3) to center (4,5) is d = sqrt((4 - -2)^2 + (5 - -3)^2) = sqrt(6^2 + 8^2) = 10. The angle between tangents is 120, so the angle between the line connecting the center to P and the tangent is 60. Let r be the radius. sin(60) = r/d = r/10. r = 10 * sqrt(3)/2 = 5*sqrt(3). The length of the tangent L = sqrt(d^2 - r^2) = sqrt(100 - 75) = sqrt(25) = 5.

Multiple choice

The length of tangent from any point in the circle $x^2+y^2+4x-6y-12=0$ to the circle $x^2+y^2+4x-6y+4=0$ is?
  1. $2$
  2. $16$
  3. $8$
  4. $4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The length of the tangent from a point (x1, y1) to a circle S=0 is sqrt(S(x1, y1)). Here, the point is on the first circle, so S1(x1, y1) = 0. The length of the tangent to the second circle S2=0 is sqrt(S2(x1, y1)). S2 = x^2+y^2+4x-6y+4 = (x^2+y^2+4x-6y-12) + 16 = 0 + 16 = 16. The length is sqrt(16) = 4.

Multiple choice

The length of tangents from P(1, -1) and Q(3,3) to a circle are $\sqrt{2}and \sqrt{6}$ respectively, then the length of tangent from R(-2,-7) to the same circle is
  1. $\sqrt{41}$
  2. $\sqrt{51}$
  3. $\sqrt{6}$
  4. $\sqrt{71}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Length of tangent from (x1, y1) to circle x^2 + y^2 + 2gx + 2fy + c = 0 is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). Let the circle be x^2 + y^2 + 2gx + 2fy + c = 0. Tangent from P(1,-1) is sqrt(1+1+2g-2f+c) = sqrt(2), so 2+2g-2f+c = 2 => 2g-2f+c = 0. Tangent from Q(3,3) is sqrt(9+9+6g+6f+c) = sqrt(6), so 18+6g+6f+c = 6 => 6g+6f+c = -12. Tangent from R(-2,-7) is sqrt(4+49-4g-14f+c) = sqrt(53-4g-14f+c). Solving the system for g, f, c is underdetermined, but the power of the point can be found.

Multiple choice

A circle with centre $(a, b)$ passes through the origin. The equation of the tangent to the circle at the origin is
  1. $ax - by = 0$
  2. $ax + by = 0$
  3. $bx- ay = 0$
  4. $bx + ay = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Circle center (a, b) passing through origin (0,0). Radius squared = a^2 + b^2. Equation: (x-a)^2 + (y-b)^2 = a^2 + b^2 => x^2 - 2ax + a^2 + y^2 - 2by + b^2 = a^2 + b^2 => x^2 + y^2 - 2ax - 2by = 0. Tangent at origin: set linear terms to 0, so -2ax - 2by = 0 => ax + by = 0.

Multiple choice

The circles of radii a and b touch externally. AB is a common tangent. The radius of the circle which touches the line AB and the two circles externally is
  1. $\sqrt { a b }$
  2. $\frac { a + b } { 2 }$
  3. $\frac { 2 a b } { a + b }$
  4. $\frac { a b } { ( \sqrt { a } + \sqrt { b } ) ^ { 2 } }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two circles of radii a and b touching externally, the radius r of a circle touching both and their common tangent is given by the formula 1/sqrt(r) = 1/sqrt(a) + 1/sqrt(b). Solving for r: 1/sqrt(r) = (sqrt(b) + sqrt(a)) / sqrt(ab), so sqrt(r) = sqrt(ab) / (sqrt(a) + sqrt(b)). Squaring both sides gives r = ab / (sqrt(a) + sqrt(b))^2.

Multiple choice

The lines 3x - 4y + 4 =0 and 6x -8y -7 =0 are tangent to the same circle the radius of this circle is

  1. 1

  2. 2

  3. 1/4

  4. 3/4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The lines are 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0. Rewrite the second as 3x - 4y - 3.5 = 0. The distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1 - c2| / sqrt(a^2 + b^2). Distance = |4 - (-3.5)| / sqrt(3^2 + 4^2) = 7.5 / 5 = 1.5. The diameter is 1.5, so the radius is 0.75 or 3/4.