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 locus of the point of intersection of the tangents at the ends of a chord of a circle $\mathrm{x}^{2}+ \mathrm{y}^{2}=\mathrm{a}^{2}$ which touches the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-2\mathrm{a}\mathrm{x}=0$
  1. $\mathrm{y}^{2}=\mathrm{a} (a-2x)$
  2. $\mathrm{x}^{2}=\mathrm{a}(\mathrm{a}-2\mathrm{y})$
  3. $\mathrm{x}^{2}-\mathrm{y}^{2}=(\mathrm{x}-\mathrm{a})^{2}$
  4. $\mathrm{x}^{2}+\mathrm{y}^{2}=(\mathrm{y}-\mathrm{a})^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord of x^2 + y^2 = a^2 touching x^2 + y^2 - 2ax = 0 is a standard geometry problem. The locus of the intersection of tangents at the ends of a chord is the polar of the intersection point. The result is y^2 = a(a-2x).

Multiple choice

The chord of contact of tangents from three points $A, B, C$ to the circle $x^2 + y^2 = a^2$ are concurrent, then A, B, C will

  1. Be concyclic

  2. Be collinear

  3. Form the vertices of a triangle

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

The chord of contact of a point (x1, y1) with respect to x^2 + y^2 = a^2 is xx1 + yy1 = a^2. If the chords of contact for three points are concurrent, the points themselves must lie on a line (the polar line of the point of concurrency).

Multiple choice

The chord of contact of pair of tangents drawn from a point $P$ to the circle $x^{2}+y^{2}=1$ passes through a fixed point $\left(1/2,1/4\right)$. Then $P$lies
  1. On the line $x+2y=4$
  2. On the line $2x+y=4$
  3. On the line $x-y=4$

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

The chord of contact of P(x1, y1) to x^2 + y^2 = 1 is xx1 + yy1 = 1. It passes through (1/2, 1/4), so (1/2)x1 + (1/4)y1 = 1. Multiplying by 4 gives 2x1 + y1 = 4. Thus P(x1, y1) lies on the line 2x + y = 4.

Multiple choice

The length of chord of contact of the point $( 3,6 )$ with respect to the circle $x ^ { 2 } + y ^ { 2 } = 10$ is
  1. $\dfrac { 2 \sqrt { 70 } } { 3 }$
  2. $6 \sqrt { 5 }$
  3. $\sqrt { 5 }$
  4. $\dfrac { 12 } { \sqrt { 5 } }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Chord of contact for (x1, y1) with circle x^2+y^2=r^2 is xx1+yy1=r^2. Here 3x+6y=10. Distance from center (0,0) to chord = |0+0-10| / sqrt(3^2+6^2) = 10 / sqrt(45) = 10 / 3sqrt(5) = 2sqrt(5)/3. Length of chord = 2 * sqrt(r^2 - d^2) = 2 * sqrt(10 - 20/9) = 2 * sqrt(70/9) = 2sqrt(70)/3.

Multiple choice

The chord of contact of tangent from a point $P$ to a circle passes through $Q$. If ${ l }{ 1 }$ and ${ l }{ 2 }$ are the lengths of the tangents from $P$ and $Q$ to the circle, then $PQ$ is equal to $\sqrt { { { l }{ 1 } }^{ 2 }+{ { l }{ 2 } }^{ 2 } } $

  1. Assertion is true, Reason is true and reason is correct explanation for Statement-1.

  2. Assertion is true, Reason is true and Reason is NOT correct explanation for Assertion.

  3. Assertion is true, Reason is false

  4. Assertion is false, Reason is true
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

If the area of a circle inscribed in an equilateral triangle is $4\pi cm^{2}$. then what is the area of the triangle?
  1. $\displaystyle 12\sqrt{3}cm^{2}$
  2. $\displaystyle 9\sqrt{3}cm^{2}$
  3. $\displaystyle 8\sqrt{3}cm^{2}$
  4. $\displaystyle 18cm^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The radius r of the inscribed circle is found from pi*r^2 = 4*pi, so r = 2. For an equilateral triangle, the inradius r is related to the side length s by r = s / (2*sqrt(3)), so s = 2*sqrt(3)*r = 4*sqrt(3). The area of the triangle is (sqrt(3)/4) * s^2 = (sqrt(3)/4) * (16 * 3) = 12*sqrt(3).

Multiple choice

What is the area of a right angled triangle if the radius of its circumcircle is 3 cm and altitude drawn to the hypotenuse is 2 cm?
  1. $\displaystyle 4\sqrt{3}cm^{2}$
  2. 6 $\displaystyle cm^{2}$
  3. 7 $\displaystyle cm^{2}$

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Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a right triangle, the circumradius R = hypotenuse / 2. So hypotenuse = 2 * 3 = 6 cm. The area of a triangle is 1/2 * base * height. Using the hypotenuse as the base and the given altitude as the height, Area = 1/2 * 6 * 2 = 6 cm^2.