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

$\displaystyle C_1$ is a circle on the major axis and $\displaystyle C_2$ is a circle on the minor axis of the ellipse $\displaystyle P : \frac {x^2}{25} + \frac {y^2}{16} = 1$ as diameters. If $l$ is the length of the tangent from any point on the circle $\displaystyle C_1$ to the ellipse $P$, then
  1. $\displaystyle l < \dfrac23$
  2. $\displaystyle l = 3$
  3. $\displaystyle l \leq \dfrac34$
  4. $\displaystyle l = \dfrac34$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle C1 has the major axis (length 10) as its diameter, so its radius is 5. The ellipse P is x^2/25 + y^2/16 = 1. The length of a tangent from a point (x1, y1) to an ellipse is given by the square root of the power of the point with respect to the ellipse. For points on the circle x^2 + y^2 = 25, the calculation shows the length l satisfies l <= 3/4.

Multiple choice

The locus of the foot of the perpendicular from the origin to chords of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4\mathrm{x}-6\mathrm{y}-3=0$ which substend a right angle at the origin, is
  1. $2(\mathrm{x}^{2}+\mathrm{y}^{2})-4\mathrm{x}-6\mathrm{y}-3=0$
  2. $\mathrm{x}^{2}+\mathrm{y}^{2}-4\mathrm{x}-6\mathrm{y}-3=0$
  3. $2(\mathrm{x}^{2}+\mathrm{y}^{2})+4\mathrm{x}+6\mathrm{y}-3=0$
  4. $2(\mathrm{x}^{2}+\mathrm{y}^{2})+4\mathrm{x}+6\mathrm{y}+3=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord subtends 90 degrees at the origin. If the chord equation is lx + my = 1, the condition for 90 degrees is a^2 + b^2 + 2g(l) + 2f(m) + c = 0. The locus of the foot of the perpendicular (h, k) from origin to the chord is h^2 + k^2 = hx + ky. Substituting into the circle equation leads to the result.

Multiple choice

The locus of midpoints of the chords of contact of $x^2 + y^2 = 2$ from the points on the line $3x + 4y = 10$ is a circle with centre $P$. If $O$ be the origin, then $OP$ is equal to
  1. $2$
  2. $3$
  3. $\dfrac{1}{2}$
  4. $\dfrac{1}{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Chord of contact from (x1, y1) to x^2+y^2=2 is xx1+yy1=2. The midpoint (h, k) of this chord satisfies h = 2x1/(x1^2+y1^2) and k = 2y1/(x1^2+y1^2). Since (x1, y1) is on 3x+4y=10, 3(2h/(h^2+k^2)) + 4(2k/(h^2+k^2)) = 10. This simplifies to 6h+8k = 10(h^2+k^2), or h^2+k^2 - 0.6h - 0.8k = 0. The center P is (0.3, 0.4). OP = sqrt(0.3^2 + 0.4^2) = 0.5.

Multiple choice

The locus of the middle point of the chord of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-2\mathrm{x}=0$ passing through the origin is
  1. $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{x}=0$
  2. $\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{y}=0$
  3. $\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{x}=0$
  4. $\mathrm{x}^{2}+\mathrm{y}^{2}-\mathrm{x}-\mathrm{y}=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the midpoint be (h, k). The chord passes through the origin (0,0) and (h,k), so its equation is y = (k/h)x. The midpoint of a chord with midpoint (h,k) in a circle x^2 + y^2 - 2x = 0 is given by T = S1, resulting in xh + yk - (x+h) = h^2 + k^2 - 2h. Substituting (0,0) into this satisfies the condition, leading to the locus x^2 + y^2 - x = 0.

Multiple choice

An equation of the chord of the circle $x^{2} + y^{2} = a^{2}$ passing through the point $(2, 3)$ farthest from the centre is
  1. $ 2x + 3y = 13$
  2. $ 3x -y = 3$
  3. $ x -2y + 4 = 0$
  4. $ x -y + 1 = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The chord farthest from the center is the one perpendicular to the radius vector at that point. The slope of the radius to (2, 3) is 3/2, so the slope of the chord is -2/3. The equation is y - 3 = -2/3(x - 2), which simplifies to 3y - 9 = -2x + 4, or 2x + 3y = 13.

Multiple choice

Find the equation of the chord of contact of the point $(1,2)$ with respect to the circle ${ x }^{ 2 }+{ y }^{ 2 }+2x+3y+1=0$
  1. $4x+7y+10=0$
  2. $4x-7y+10=0$
  3. $7x+4y+10=0$
  4. $7x-4y+10=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of the chord of contact for a circle x^2 + y^2 + 2gx + 2fy + c = 0 from point (x1, y1) is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0. Substituting (1, 2) and the circle parameters (g=1, f=1.5, c=1): x(1) + y(2) + 1(x + 1) + 1.5(y + 2) + 1 = 0. This simplifies to x + 2y + x + 1 + 1.5y + 3 + 1 = 0, or 2x + 3.5y + 5 = 0. Multiplying by 2 gives 4x + 7y + 10 = 0.

Multiple choice

Let $\mathrm{A}\mathrm{B}$ be a chord of the circle $x^{2}+y^{2}=r^{2}$ subtending a right angle at the centre. Then, the locus of the centroid of the triangle $P\mathrm{A}\mathrm{B}$ as $\mathrm{P}$ moves on the circle is

  1. A parabola

  2. A circle

  3. An ellipse

  4. A pair of straight line
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let A=(r,0) and B=(0,r). Centroid G = ((r+x_p)/3, (r+y_p)/3). Since x_p^2 + y_p^2 = r^2, substituting x_p = 3x - r and y_p = 3y - r gives (3x-r)^2 + (3y-r)^2 = r^2, which is the equation of a circle.

Multiple choice

The chord of contact of the pair of tangents drawn from any point on the line $3x + y = 5$ to the circle $\displaystyle x^{2}+y^{2}=4 $ passes through
  1. $\displaystyle \left ( \frac{1}{5},\frac{2}{7} \right ) $
  2. $\displaystyle \left ( \frac{12}{7},\frac{4}{7} \right ) $
  3. $\displaystyle \left ( \frac{12}{5},\frac{4}{5} \right ) $
  4. $(1, 1)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The chord of contact of a point (h, k) with respect to x^2 + y^2 = 4 is hx + ky = 4. Since (h, k) lies on 3x + y = 5, we have k = 5 - 3h. Substituting this into the chord equation gives hx + (5 - 3h)y = 4, or 5y - 4 + h(x - 3y) = 0. This line passes through the intersection of 5y - 4 = 0 and x - 3y = 0, which is y = 4/5 and x = 3(4/5) = 12/5.

Multiple choice

Locus of mid points of the chords of the parabola $y^{2}=4ax$ which touch the circle $x^{2}+y^{2}=a^{2}$ is
  1. $(y^{2}-2a\mathrm{x})^{2}=a^{4}(y^{2}+4a^{2})$
  2. $(y^{2}-2a\mathrm{x})^{2}=a^{2}(y^{2}+4a^{2})$
  3. $(y^{2}-2a\mathrm{x})^{2}=2a^{4}(y^{2}+4a^{2})$
  4. $(y^{2}-2a\mathrm{x})^{2}=4a^{2}(y^{2}+4a^{2})$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The locus of the midpoint (h, k) of a chord of y^2=4ax is given by T=S1, which is ky - 2a(x+h) = k^2 - 4ah. This simplifies to ky = 2ax + k^2 - 2ah. The distance from the origin to this line must equal the radius 'a' of the circle x^2+y^2=a^2. Using the distance formula |k^2 - 2ah| / sqrt(k^2 + 4a^2) = a leads to the given equation.

Multiple choice

A chord of a circle of radius $10$ cm subtends a right angle at its centre. The length of the chord (in cm) is
  1. $\displaystyle 5\sqrt { 2 } $
  2. $\displaystyle 10\sqrt { 2 } $
  3. $\displaystyle \frac { 5 }{ \sqrt { 2 } } $
  4. $\displaystyle 10\sqrt { 3 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A chord subtending a right angle at the center forms an isosceles right triangle with the two radii. Using the Pythagorean theorem, the chord length is sqrt(10^2 + 10^2) = sqrt(200) = 10*sqrt(2).

Multiple choice

lf the circles $(\mathrm{x}-\mathrm{a})^{2}+(\mathrm{y}-\mathrm{b})^{2}=\mathrm{r}^{2},\ (\mathrm{x}-\mathrm{b})^{2}+(\mathrm{y}- \mathrm{a})^{2}=\mathrm{r}^{2}$ have three common tangents, then which of the following relation follows:
  1. $(a-b)^{2}$ $=2\mathrm{r}^{2}$
  2. $(\mathrm{a}+\mathrm{b})^{2}=2\mathrm{r}^{2}$
  3. $\mathrm{a}^{2}+\mathrm{b}^{2}=2\mathrm{r}^{2}$
  4. $\mathrm{a}^{2}+\mathrm{b}^{2}=\mathrm{r}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For two circles with equal radii r to have three common tangents, they must be externally tangent. The distance between centers (a, b) and (b, a) must equal 2r. Distance = sqrt((a-b)^2 + (b-a)^2) = sqrt(2(a-b)^2) = |a-b|*sqrt(2). Setting this to 2r and squaring gives 2(a-b)^2 = 4r^2, or (a-b)^2 = 2r^2.