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 locus of the middle point of the chords of contact of tangents to $x^{2}-y^{2}=a^{2}$ from points on the auxiliary circle is of the form $px^{4}+qy^{2}x^{2}+py^{4}-r(x^{2}+y^{2})=0$ then $r=$
  1. $a^{2}$
  2. $-a^{2}$
  3. $a$
  4. $-a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord of contact from (x1, y1) to x^2 - y^2 = a^2 is xx1 - yy1 = a^2. The midpoint of this chord is (h, k). The condition for the chord of contact to have a midpoint (h, k) is xh - yk = h^2 - k^2. Comparing coefficients leads to r = a^2.

Multiple choice

The equation of the circle with the chord $y=2x$ of the circle $x^{2}+y^{2}-10x=0$ as its diameter is
  1. $x^{2}+y^{2}-2x-4y-5=0$
  2. $x^{2}+y^{2}=2x+4y$
  3. $x^{2}+y^{2}=4x+2y$

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

The circle is x^2 + y^2 - 10x = 0. The intersection of this circle and the line y = 2x gives the endpoints of the diameter. Substituting y = 2x into the circle equation: x^2 + (2x)^2 - 10x = 0, so 5x^2 - 10x = 0, meaning x = 0 or x = 2. The points are (0,0) and (2,4). The equation of a circle with diameter endpoints (x1, y1) and (x2, y2) is (x-x1)(x-x2) + (y-y1)(y-y2) = 0. This gives (x-0)(x-2) + (y-0)(y-4) = 0, which is x^2 - 2x + y^2 - 4y = 0, or x^2 + y^2 = 2x + 4y.

Multiple choice

The radii of two circles are $2$ units and $3$ units. lf the radical axis of the circles cuts one of the common tangents of the circle in $\mathrm{P}$ then ratio in which $\mathrm{P}$ divides the common tangent is
  1. $2 : 3$
  2. $3 : 2$
  3. $4 : 9$
  4. $1 : 1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two circles, the radical axis is the locus of points from which the tangents to the two circles are equal in length. For any point P on the common tangent, the lengths of the tangents to both circles are equal. By the property of the radical axis and common tangents, the point P bisects the common tangent segment between the points of contact.

Multiple choice

If the centroid of tetrahedron $OABC$ where $A,B,C$ are given by $(a,2,3), (1,b,2)$ and $(2,1,c)$ respectively is $(1,2,-2)$, then distance of $P(a,b,c)$ from origin is
  1. $\sqrt{195}$
  2. $\sqrt{14}$
  3. $\sqrt{\dfrac{107}{14}}$
  4. $\sqrt{13}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The centroid of a tetrahedron with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3), (x4,y4,z4) is the average of the coordinates. Given O(0,0,0), A(a,2,3), B(1,b,2), C(2,1,c), the centroid is ((a+1+2)/4, (2+b+1)/4, (3+2+c)/4) = (1,2,-2). Solving gives a=1, b=5, c=-13. The distance of P(1,5,-13) from origin is sqrt(1^2 + 5^2 + (-13)^2) = sqrt(1+25+169) = sqrt(195).

Multiple choice

The locus of the middle points of the chords of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4\mathrm{x}-2\mathrm{y}-40=0$ which make an angle $\displaystyle \tan^{-1}\frac{4}{3}$ with the $\mathrm{x}$-axis is
  1. $4x-3y =5$
  2. $3\mathrm{x}+4\mathrm{y}=10$
  3. $4\mathrm{x}+3\mathrm{y}=11$
  4. $3\mathrm{x}-4\mathrm{y}=1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The center of the circle is (2, 1). The slope of the chord is tan(tan^-1(4/3)) = 4/3. The slope of the line connecting the center to the midpoint of the chord is the negative reciprocal of the chord's slope, which is -3/4. The equation of the line passing through (2, 1) with slope -3/4 is y - 1 = -3/4(x - 2) => 4y - 4 = -3x + 6 => 3x + 4y = 10.

Multiple choice

If the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2\mathrm{g}\mathrm{x}+2\mathrm{f}\mathrm{y}+\mathrm{c}=0$ bisects the circumference of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2\mathrm{g}{1} \mathrm{x}+2\mathrm{f}{1}\mathrm{y}+\mathrm{c}_{1}=0$, then the length of the common chord of the circles is
  1. $2\sqrt{g_{1}^{2}+f_{1}^{2}-c_{1}}$
  2. $\sqrt{g_{1}^{2}+f_{1}^{2}-c_{1}}$
  3. $\sqrt{g^{2}+f^{2}-c}$
  4. $2\sqrt{g^{2}+f^{2}-c}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a circle bisects the circumference of another, the common chord is the diameter of the bisected circle. The equation of the common chord is found by subtracting the two circle equations: 2(g-g1)x + 2(f-f1)y + (c-c1) = 0. The length of the diameter of the second circle is 2 * sqrt(g1^2 + f1^2 - c1).

Multiple choice

The range of value of $a$ for which the line $y+x=0$ bisects two chords drawn from a point $\displaystyle \left( \frac { 1+\sqrt { 2 } a }{ 2 } ,\frac { 1-\sqrt { 2 } a }{ 2 } \right) $ to the circle $2{ x }^{ 2 }+2{ y }^{ 2 }-\left( 1+\sqrt { 2 } a \right) x-\left( 1-\sqrt { 2 } a \right) y=0$ is
  1. $\left( -\infty ,-2 \right) \cup \left( 2,\infty  \right) $
  2. $\left( -2,2 \right) $
  3. $\left( 2,\infty  \right) $

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

The coordinates of the middle point of the chord which the circle ${ x }^{ 2 }+{ y }^{ 2 }+4x-2y-3=0$ cuts off on the line $y=x+2$, are
  1. $\displaystyle \left( -\frac { 3 }{ 2 } ,\frac { 1 }{ 2 }  \right) $
  2. $\displaystyle \left( \frac { 3 }{ 2 } ,\frac { 1 }{ 2 }  \right) $
  3. $\displaystyle \left( -\frac { 3 }{ 2 } ,-\frac { 1 }{ 2 }  \right) $
  4. $\displaystyle \left( \frac { 3 }{ 2 } ,-\frac { 1 }{ 2 }  \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The midpoint of a chord with equation y=mx+c cut by a circle x^2+y^2+2gx+2fy+c=0 is found by substituting the line into the circle equation or using the formula for the chord with a given midpoint. The result is (-3/2, 1/2).

Multiple choice

The distance from the origin to the centroid of the tetrahedron formed by the points $(0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c)$ is:
  1. $\displaystyle \dfrac{\sqrt{a+{b}+{c}}}{4}$
  2. $\displaystyle \dfrac{\sqrt{{a}+{b}+{c}}}{3}$
  3. $\displaystyle \dfrac{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}{16}$
  4. $\displaystyle \dfrac{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The centroid of a tetrahedron with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3), (x4,y4,z4) is ((x1+x2+x3+x4)/4, (y1+y2+y3+y4)/4, (z1+z2+z3+z4)/4). Here, vertices are (0,0,0), (a,0,0), (0,b,0), (0,0,c). Centroid = (a/4, b/4, c/4). Distance from origin = sqrt((a/4)^2 + (b/4)^2 + (c/4)^2) = sqrt(a^2 + b^2 + c^2) / 4.

Multiple choice

A chord of length $L$ cm is drawn in a circle of radius $R$ cm. The distance of the chord from the center of the circle is
  1. $\displaystyle \sqrt{L^{2}}-\frac{R^{2}}{4}$
  2. $\displaystyle \sqrt{\frac{4R^{2}-L^{2}}{4}}$
  3. $\displaystyle \beta $
  4. $\displaystyle \sqrt{R^{2}}+L^{2}-RL$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a circle, the radius, half-chord, and distance from center form a right triangle. R^2 = d^2 + (L/2)^2. Thus, d^2 = R^2 - L^2/4 = (4R^2 - L^2)/4. d = sqrt((4R^2 - L^2)/4).

Multiple choice

The length of the tangents from a point A to a circle of radius $3$ cm is $4$ cm. The distance (in cm) of A from the center of the circle is:
  1. $\sqrt{7}$
  2. $7$
  3. $5$
  4. $25$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. This forms a right triangle with the radius (3) and the tangent (4) as legs. The distance from the center to point A is the hypotenuse: sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.