List of top Mathematics Questions on Geometry asked in JEE Main

A rectangle is formed by the lines $ x = 0,\ y = 0,\ x = 3,\ y = 4 $. A line perpendicular to $ 3x + 4y + 6 = 0 $ divides the rectangle into two equal parts. Then the distance of the line from the point $ \left(-1,\frac{3}{2}\right) $ is:

If $ L_1 $ and $ L_2 $ are two parallel lines and $ \triangle ABC $ is an equilateral triangle as shown in the figure, then the area of triangle $ ABC $ is:

If \[ \lim_{x \to 0} \frac{e^{a(x-1)} + 2\cos(bx) + e^{-x}(c - 1)}{x \cos x - \ln(1 + x)} = 2, \] Then the value of $ a^2 + b^2 + c^2 $ is:

If $ P(10, 2\sqrt{15}) $ lies on the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ and the length of the latus rectum is 8, then the square of the area of $ \Delta PS_1S_2 $ is [where $ S_1 $ and $ S_2 $ are the foci of the hyperbola].

If $ \cos(\alpha + \beta) = -\frac{1}{10} $ and $ \sin(\alpha - \beta) = \frac{3}{8} $, where \[ 0<\alpha<\frac{\pi}{3} \quad \text{and} \quad 0<\beta<\frac{\pi}{4}, \] and \[ \tan(2\alpha) = \frac{3\left(1 - \sqrt{55}\right)}{\sqrt{11} \left(s + \sqrt{5}\right)}, \] and $ r, s \in \mathbb{N} $, then $ r^2 + s $ is:

If $ O $ is the vertex of the parabola $ x^2 = 4y $, $ Q $ is a point on the parabola. If $ C $ is the locus of a point which divides $ OQ $ in the ratio $ 2:3 $, then the equation of the chord of $ C $ which is bisected at the point $ (1,2) $ is:

The locus of the point of intersection of tangents drawn to the circle \[ (x - 2)^2 + (y - 3)^2 = 16, \] which subtends an angle of $120^\circ$, is:

The line $y = x + 1$ intersects the ellipse \[ \frac{x^2}{2} + \frac{y^2}{1} = 1 \] at points $A$ and $B$. Find the angle subtended by the segment $AB$ at the centre of the ellipse.

Let a circle $ C $ with radius $ r $ passes through four distinct points $ (0, 0), (k, 3k), (2, 3), (-1, 5) $, such that $ k \neq 0 $, then $ (10k + 2r^2) $ is equal to:

Let $ ABCD $ be a tetrahedron such that the edges $ AB $, $ AC $, and $ AD $ are mutually perpendicular. Let the areas of the triangles $ ABC $, $ ACD $, and $ ADB $ be 5, 6, and 7 square units respectively. Then the area (in square units) of the $ \triangle BCD $ is equal to:

Let for two distinct values of $ p $, the lines $ y = x + p $ touch the ellipse $ E: \frac{x^2}{4} + \frac{y^2}{9} = 1 $ at the points $ A $ and $ B $. Let the line $ y = x $ intersect $ E $ at the points $ C $ and $ D $. Then the area of the quadrilateral $ ABCD $ is equal to:

Let $ A(4, -2), B(1, 1) $ and $ C(9, -3) $ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.

Let the three sides of a triangle ABC be given by the vectors $$ 2\hat{i} - \hat{j} + \hat{k}, \quad \hat{i} - 3\hat{j} - 5\hat{k}, \quad \text{and} \quad 3\hat{i} - 4\hat{j} - 4\hat{k}. $$ Let G be the centroid of the triangle ABC. Then $$ 6 \left( |\vec{AG}|^2 + |\vec{BG}|^2 + |\vec{CG}|^2 \right) \text{ is equal to}$$ _______

The center of a circle $ C $ is at the center of the ellipse $ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, where $ a>b $. Let $ C $ pass through the foci $ F_1 $ and $ F_2 $ of $ E $ such that the circle $ C $ and the ellipse $ E $ intersect at four points. Let $ P $ be one of these four points. If the area of the triangle $ PF_1F_2 $ is 30 and the length of the major axis of $ E $ is 17, then the distance between the foci of $ E $ is:

Let the sum of the focal distances of the point $ P(4, 3) $ on the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be $ 8\sqrt{\frac{5}{3}} $. If for $ H $, the length of the latus rectum is $ \ell $ and the product of the focal distances of the point $ P $ is $ m $, then $ 9\ell^2 + 6m $ is equal to:

The axis of a parabola is the line $ y = x $ and its vertex and focus are in the first quadrant at distances $ \sqrt{2} $ and $ 2\sqrt{2} $ units from the origin, respectively. If the point $ (1, k) $ lies on the parabola, then a possible value of $ k $ is:

Let the values of $ p $, for which the shortest distance between the lines $ \frac{x + 1}{3} = \frac{y}{4} = \frac{z}{5} $ and $ \vec{r} = (p \hat{i} + 2 \hat{j} + \hat{k}) + \lambda (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) $ is $ \frac{1}{\sqrt{6}} $, be $ a, b $, where $ a<b $. Then the length of the latus rectum of the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ is:

Let $ r $ be the radius of the circle, which touches the $ x $-axis at point $ (a, 0) $, $ a < 0 $ and the parabola $ y^2 = 9x $ at the point $ (4, 6) $. Then $ r $ is equal to:

The distance of the point $ (7, 10, 11) $ from the line $ \frac{x-9}{2} = \frac{y-13}{3} = \frac{z-17}{6} $ along the line is:

From the top A of a vertical wall AB of height 30 m, the angles of depression of the top P and bottom Q of a vertical tower PQ are 15$^\circ$ and 60$^\circ$ respectively. B and Q are on the same horizontal level. If C is a point on AB such that CB = 15 m, then the area (in m$^2$) of the quadrilateral BCPQ is equal to :

Let the equation of the plane, that passes through the point $(1, 4, -3)$ and contains the line of intersection of the planes $3x - 2y + 4z - 7 = 0$ and $x + 5y - 2z + 9 = 0$, be $\alpha x + \beta y + \gamma z + 3 = 0$, then $\alpha + \beta + \gamma$ is equal to :

If the variable line $3x + 4y = \alpha$ lies between the two circles $(x-1)^2 + (y-1)^2 = 1$ and $(x-9)^2 + (y-1)^2 = 4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is _____________