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 length of the tangent from (h, k) to the circle $x^2 \, + \, y^2 \, = \, 16$ is twice the length of the tangent from the same point to the circle $x^2 \, + \, y^2 \, + 2x \, + \, 2y \, = \, 0$, then
  1. $h^2 \, + \, k^2 \, + \, 4h \,+ \, 4k \, + \, 16 \, = \, 0$
  2. $h^2 \, + \, k^2 \, + \, 3h \,+ \, 3k \, = \, 0$
  3. $3h^2 \, + \, 3k^2 \, + \, 8h \,+ \, 8k \, + \, 16 \, = \, 0$
  4. $3h^2 \, + \, 3k^2 \, + \, 4h \,+ \, 4k \, + \, 16 \, = \, 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Length of tangent from (h,k) to x^2+y^2-16=0 is sqrt(h^2+k^2-16). To x^2+y^2+2x+2y=0 is sqrt(h^2+k^2+2h+2k). Given sqrt(h^2+k^2-16) = 2 * sqrt(h^2+k^2+2h+2k). Squaring: h^2+k^2-16 = 4(h^2+k^2+2h+2k) = 4h^2+4k^2+8h+8k. Rearranging: 3h^2+3k^2+8h+8k+16=0.

Multiple choice

If the lines $3x+4y-14=0$ and $6x+8y+7=0$ are both tangents to a circle, then its radius is
  1. $7$
  2. $\dfrac{7}{2}$
  3. $\dfrac{7}{4}$
  4. $\dfrac{7}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1 - c2| / sqrt(a^2 + b^2). Here, 6x + 8y + 7 = 0 is 3x + 4y + 3.5 = 0. Distance = | -14 - 3.5 | / sqrt(3^2 + 4^2) = 17.5 / 5 = 3.5. The radius is half the distance between tangents, so 3.5 / 2 = 1.75 or 7/4.

Multiple choice

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is
  1. $20\left(x^2 + y^2 \right) - 36x + 45y = 0 $
  2. $20\left(x^2 + y^2 \right) + 36x - 45y = 0 $
  3. $20\left(x^2 + y^2 \right) - 20x + 45y = 0 $
  4. $20\left(x^2 + y^2 \right) + 20x - 45y = 0 $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord of contact of a point (h, k) with respect to x^2 + y^2 = r^2 is hx + ky = r^2. Given the point lies on 4x - 5y = 20, we have 4h - 5k = 20, or 4h = 5k + 20. The mid-point (x, y) of the chord of contact satisfies x = h(r^2)/(h^2 + k^2) and y = k(r^2)/(h^2 + k^2). Substituting r^2 = 9 and the relation between h and k leads to the resulting locus.

Multiple choice

Let $S$ be the focus of the parabola $y^2 = 8x$ and let $PQ$ be the common chord of the circle $x^2 +y^2 - 2x - 4y = 0$ and the given parabola. The area of the $\triangle PQS$ is
  1. $4$
  2. $5$
  3. $6$
  4. $7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The parabola is y^2 = 8x (focus S at (2,0)). The circle is x^2 + y^2 - 2x - 4y = 0. Substituting y^2 = 8x into the circle equation: x^2 + 8x - 2x - 4y = 0, x^2 + 6x - 4y = 0. Solving for intersection points P and Q gives the chord. The area is calculated using the base PQ and height from S.

Multiple choice

The point P is on circle $\displaystyle { x }^{ 2 }+{ y }^{ 2 }=36$ and tangents are drawn to the circle $\displaystyle { x }^{ 2 }+{ y }^{ 2 }=18$ from P. The angle between the tangent lines is
  1. $\displaystyle { 30 }^{ \circ }$
  2. $\displaystyle { 60 }^{ \circ }$
  3. $\displaystyle { 45 }^{ \circ }$
  4. $\displaystyle { 90 }^{ \circ }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The point P is on x^2 + y^2 = 36 (radius 6). The tangents are to x^2 + y^2 = 18 (radius sqrt(18)). In the right triangle formed by the center, the point of tangency, and P, the hypotenuse is 6 and one leg is sqrt(18). sin(theta/2) = sqrt(18)/6 = sqrt(2)/2. Thus theta/2 = 45 degrees, and theta = 90 degrees.

Multiple choice

The lengths of the tangent drawn from any point on the circle $15x^{2} + 15y^{2} - 48x + 64y = 0$ to the two circles $5x^{2} + 5y^{2} - 24x + 32y + 75 = 0$ and $5x^{2} + 5y^{2} - 48x + 64y + 300 = 0$ are in the ratio of
  1. $1 : 2$
  2. $2 : 3$
  3. $3 : 4$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The power of a point (x, y) with respect to a circle x^2+y^2+2gx+2fy+c=0 is x^2+y^2+2gx+2fy+c. The circle 15x^2+15y^2-48x+64y=0 is x^2+y^2-3.2x+4.26y=0. The two circles are C1: x^2+y^2-4.8x+6.4y+15=0 and C2: x^2+y^2-9.6x+12.8y+60=0. The ratio of the powers of any point on the first circle to C1 and C2 is constant. Calculating the ratio of the constants or radii shows the ratio of lengths is 1:2.

Multiple choice

The length of the chord $x + y = 3$ intercepted by the circle $x^{2} + y^{2} - 2x - 2y - 2 = 0$ is
  1. $\dfrac {7}{2}$
  2. $\dfrac {3\sqrt {3}}{2}$
  3. $\sqrt {14}$
  4. $\dfrac {\sqrt {7}}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle center is (1, 1) and radius is sqrt(1^2 + 1^2 + 2) = 2. The distance from (1, 1) to x + y - 3 = 0 is |1 + 1 - 3| / sqrt(1^2 + 1^2) = 1/sqrt(2). The half-length of the chord is sqrt(r^2 - d^2) = sqrt(4 - 1/2) = sqrt(7/2). The total length is 2 * sqrt(7/2) = sqrt(14).

Multiple choice

Equation of the chord of the circle $x^{2} + y^{2} + 4x - 6y - 9 = 0$ bisected at $(0, 1)$ is
  1. $y - 1 = x$
  2. $y + 1 = x$
  3. $y + 1 = 2x$
  4. $y - 1 = 3x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a chord of a circle bisected at (x1, y1) is T = S1. Here, T = x(0) + y(1) + 2(x+0) - 3(y+1) - 9 = y + 2x - 3y - 3 - 9 = 2x - 2y - 12. S1 = 0^2 + 1^2 + 4(0) - 6(1) - 9 = 1 - 6 - 9 = -14. Setting 2x - 2y - 12 = -14 gives 2x - 2y = -2, or x - y = -1, which simplifies to y - 1 = x.

Multiple choice

The length of the tangent drawn from a point on the circle $x^{2} + y^{2} - 4x + 6y - 4 = 0$ to the circle $x^{2} + y^{2} - 4x + 6y = 0$ is
  1. $8$
  2. $4$
  3. $2$

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

Circle 1: x^2 + y^2 - 4x + 6y - 4 = 0. Circle 2: x^2 + y^2 - 4x + 6y = 0. These are concentric circles with centers at (2, -3). The radius of circle 1 is sqrt(4 + 9 + 4) = sqrt(17). The radius of circle 2 is sqrt(4 + 9) = sqrt(13). The length of the tangent from a point on circle 1 to circle 2 is sqrt(R^2 - d^2), where d is the distance between centers (0 here) and R is the radius of the outer circle. This is not correct. The tangent from a point on C1 to C2 is sqrt(dist(point, center)^2 - r2^2). Since the point is on C1, dist^2 = r1^2 = 17. Length = sqrt(17 - 13) = sqrt(4) = 2.