List of top Mathematics Questions on Coordinate Geometry asked in JEE Advanced

Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is

Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is

Let $ S $ denote the locus of the midpoints of those chords of the parabola $ y^2 = x $, such that the area of the region enclosed between the parabola and the chord is $ \frac{4}{3} $. Let $ \mathcal{R} $ denote the region lying in the first quadrant, enclosed by the parabola $ y^2 = x $, the curve $ S $, and the lines $ x = 1 $ and $ x = 4 $.
Then which of the following statements is (are) TRUE?

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ z_1 = 1 + 2i $ and $ z_2 = 3i $ be two complex numbers, where $ i = \sqrt{-1} $. Let $$ S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|\}. $$ Then which of the following statements is (are) TRUE?

For any two points $ M $ and $ N $ in the $ XY $-plane, let $ \overrightarrow{MN} $ denote the vector from $ M $ to $ N $, and $ \vec{0} $ denote the zero vector. Let $ P, Q $, and $ R $ be three distinct points in the $ XY $-plane. Let $ S $ be a point inside the triangle $ \Delta PQR $ such that $$ \overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \vec{0}. $$ Let $ E $ and $ F $ be the mid-points of the sides $ PR $ and $ QR $, respectively. Then the value of $$ \frac{\text{length of the line segment } EF}{\text{length of the line segment } ES} $$ is __________.

Let $S=\left\{(x,y)\in\R\times\R:x\ge0,y\ge0,y^2\le12-2x\ \text{and}\ 3y+\sqrt8\ x\le5\sqrt8\right\}$. If the area of the region S is $\alpha\sqrt2$, then α is equal to

Let A₁, B₁, C₁ be three points in the xy -plane. Suppose that the lines A₁C₁ and B₁C₁ are tangents to the curve y² = 8x at A₁ and B₁, respectively. If O = (0,0) and C₁ = −( 4, 0), then which of the following statements is (are) TRUE ?

Let the straight line y = 2x touch a circle with center (0, α), α > 0, and radius r at a point A₁. Let B₁ be the point on the circle such that the line segment A₁B₁ is a diameter of the circle. Let $α+r=5+\sqrt5$.
Match each entry in List-I to the correct entry in List-II.

List - I		List - II
(P)	α equals	(1)	(-2, 4)
(Q)	r equals	(2)	$\sqrt5$
(R)	A₁ equals	(3)	(-2, 6)
(S)	B₁ equals	(4)	5
		(5)	(2, 4)

The correct option is

Consider a triangle $Δ$ whose two sides lie on the x-axis and the line $x + y + 1 = 0$. If the orthocenter of $Δ$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $Δ$ is;

Let $L_{1}$ and $L_{2}$ be the following straight lines
$L_{1}: \frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_{2}: \frac{x-1}{-3}-\frac{y}{-1}=\frac{z-1}{1} $
Suppose thestraight line
$L: \frac{x-\alpha}{1}=\frac{y-1}{m}=\frac{z-\gamma}{-2}$
lies in the plane containing $L_{1}$ and $L_{2}$, and passes through thepoint of intersection of $L_{1}$ and $L_{2}$. If the line $L$ bisects the acute angle between the lines $L_{1}$ and $L_{2}$, then which of the following statements is/are TRUE?