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 tangent to the circle ${x^2} + {y^2} = 5$ at the point $\left( {1, - 2} \right)$ also touches the circle ${x^2} + {y^2} - 8x + 6y + 20 = 0$ at
  1. $\left( { - 2,1} \right)$
  2. $\left( { - 3,0} \right)$
  3. $\left( { - 1, - 1} \right)$
  4. $\left( { 3, - 1} \right)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Tangent to x^2 + y^2 = 5 at (1, -2) is x - 2y = 5, or x = 2y + 5. Substitute into the second circle: (2y+5)^2 + y^2 - 8(2y+5) + 6y + 20 = 0. 4y^2 + 20y + 25 + y^2 - 16y - 40 + 6y + 20 = 0. 5y^2 + 10y + 5 = 0. y^2 + 2y + 1 = 0, so (y+1)^2 = 0, y = -1. If y = -1, x = 2(-1) + 5 = 3. Point is (3, -1).

Multiple choice

Tangents are drawn through the point $P(-2, -3)$ to touch the circle $x^{2}+y^{2}-2x-10y+1=0$. The length of line segment $PA$, $A$ being the point on the circle where the line meets the circle at coincident points, is
  1. $16$
  2. $4\sqrt{3}$
  3. $48$

  4. none of these
Reveal answer Fill a bubble to check yourself
B 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). Here, g=-1, f=-5, c=1. Point (-2, -3). Length = sqrt((-2)^2 + (-3)^2 - 2(-2) - 10(-3) + 1) = sqrt(4 + 9 + 4 + 30 + 1) = sqrt(48) = 4*sqrt(3).

Multiple choice

From the point $P (4,3)$ tangents $PA$ and $PB$ are drawn to the circle ${ x }^{ 2 }+{ y }^{ 2 }=9$ Than area of the triangle $PAB$ is-
  1. $\dfrac{192}{25}$sq units
  2. $\dfrac {16}{9}$ sq units
  3. $\dfrac { 21\sqrt { 3 } }{ 16\quad } sq\quad units$
  4. $\dfrac { 25\sqrt { 3 } }{ 9\quad } sq\quad units$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle x^2 + y^2 = r^2, the length of the tangent from (x1, y1) is sqrt(x1^2 + y1^2 - r^2). Here, length L = sqrt(4^2 + 3^2 - 3^2) = 4. The area of triangle PAB is (L^3 * r) / (L^2 + r^2) = (64 * 3) / (16 + 9) = 192 / 25.

Multiple choice

Three circles with radii $3$ cm, $4$ cm and $5$ cm touch each other externally. If $A$ is the point of intersection of tangents to these circles at !heir points of contact. then !he distance of A from !he points of contact is
  1. $\sqrt{3} $
  2. $2$
  3. $\sqrt{5}$
  4. $\sqrt{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the tangent-length relationships for three externally touching circles with radii 3, 4, and 5 cm, the required tangent distance evaluates to sqrt(5) cm. Therefore, option C matches the result.

Multiple choice

If the locus of the point of intersection of perpendicular tangents to the ellipse $\dfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\dfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$ is a circle with centre at $\left (0,0\right)$, then the radius of the circle would be
  1. $a+b$
  2. $ab$
  3. $\dfrac b a$
  4. $\sqrt { ({ a }^{ 2 }+{ b }^{ 2 }) } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The locus of the intersection of perpendicular tangents to an ellipse is its director circle, which is given by x^2 + y^2 = a^2 + b^2. The radius is sqrt(a^2 + b^2).

Multiple choice

Point $O$ is the centre of the ellipse with major axis $AB$ and minor axis $CD$. Point $F$ is one focus of the ellipse.If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is $2$, then the product $(AB)(CD)$ is equal to
  1. $65$
  2. $52$
  3. $78$

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For an ellipse, OF = ae = 6. The triangle OCF has sides OC=b, OF=ae, and CF=a. The diameter of the inscribed circle is 2, so the radius r = 1. The area of triangle OCF is rs = 1 * (a+b+ae)/2. Also, area = 0.5 * b * ae. Equating these and using a^2 = b^2 + (ae)^2 leads to the result.

Multiple choice

From the points on the circle $x^2+y^2=a^2$, tangents are drawn to the hyperbola $x^2-y^2=a^2$; the locus of the middle points of the chords of contact is
  1. $\displaystyle\frac{x^2-y^2}{a^2x^2+y^2}=a^2$
  2. ${(x^2-y^2)}^2=a^2(x^2+y^2)$
  3. ${(x^2+y^2)}^2=a^2(x^2-y^2)$
  4. $x^2+y^2=a^2(x^2-y^2)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord of contact of tangents from (x1, y1) to the hyperbola x^2 - y^2 = a^2 is xx1 - yy1 = a^2. The midpoint of this chord is (h, k), so the equation of the chord is xh - yk = h^2 - k^2. Comparing these, we find h/x1 = -k/y1 = a^2 / (h^2 - k^2). Since (x1, y1) lies on the circle x^2 + y^2 = a^2, substituting these values leads to the locus (x^2 - y^2)^2 = a^2(x^2 + y^2).

Multiple choice

Circle drawn with a focal chord of the parabola $y ^ { 2 } = 4 x + y + 4$ as diameter will always touch the line $16 x + k = 0 ,$ where $k$ equals to :
  1. $8$
  2. $16$
  3. $33$
  4. $36$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The parabola is y^2 - y = 4x + 4. Completing the square: (y - 1/2)^2 = 4x + 4 + 1/4 = 4x + 17/4 = 4(x + 17/16). The focus is at (h+a, k) = (-17/16 + 1, 1/2) = (-1/16, 1/2). A circle with a focal chord as diameter touches the directrix. The directrix is x = h - a = -17/16 - 1 = -33/16. The line is 16x + k = 0, or x = -k/16. Comparing x = -33/16 and x = -k/16, k = 33.