List of top Mathematics Questions on Parabola asked in JEE Main

Let \( L_1, L_2 \) be the lines passing through the point \( P(0, 1) \) and touching the parabola \[ 9x^2 + 12x + 18y - 14 = 0. \] Let \( Q \) and \( R \) be the points on the lines \( L_1 \) and \( L_2 \) such that the \( \Delta PQR \) is an isosceles triangle with base \( QR \). If the slopes of the lines \( QR \) are \( m_1 \) and \( m_2 \), then \( 16\left(m_1^2 + m_2^2\right) \) is equal to ______.

Let \( C \) be the circle of minimum area touching the parabola \( y = 6 - x^2 \) and the lines \( y = \sqrt{3} |x| \). Then, which one of the following points lies on the circle \( C \)?

Let the length of the focal chord \( PQ \) of the parabola \( y^2 = 12x \) be 15 units. If the distance of \( PQ \) from the origin is \( p \), then \( 10p^2 \) is equal to _____

Let \( PQ \) be a chord of the parabola \( y^2 = 12x \) and the midpoint of \( PQ \) be at \( (4, 1) \). Then, which of the following points lies on the line passing through the points \( P \) and \( Q \)?

The area (in square units) of the region described by: \[ \{(x, y) : y^2 \leq 2x, \, \text{and} \, y \geq 4x - 1\} \] is:

The area of the region enclosed by the parabola \((y - 2)^2 = x - 1\), the line \(x - 2y + 4 = 0\) and the positive coordinate axes is ______.

The area of the region enclosed by the parabolas \(y = 4 – x^2\) and \(3y = (x – 4)^2\) is in (sq. unit)?

\((α, β)\) lie on the parabola \(y^2 = 4x\) and \((α, β)\) also lie on chord with midpoint \((1,\frac 54)\) of another parabola \(x^2 = 8y\), then value of \(|(8 – β)(α – 28)|\) is

If the $x$-intercept of a focal chord of the parabola $y^2=8 x+4 y+4$ is 3 , then the length of this chord is equal to ___

Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P ( b, c), b, c \in N$, on the parabola $y^2=2$ ax and the lines $x=b, y=0$ is $16$ unit ${ }^2$, then $\displaystyle\sum_{a \in S} a$ is equal to _____

Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P ( b, c), b, c \in N$, on the parabola $y^2=2$ ax and the lines $x=b, y=0$ is $16$ unit ${ }^2$, then $\displaystyle\sum_{a \in S} a$ is equal to _____

Let PQ be a focal chord of the parabola y² = 36x of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM:MQ = 3:1. Then which of the following points does \(\underline{NOT}\) lie on the line passing through M and perpendicular to the line PQ? 

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$ Then $S=\left\{x \in R : \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$ :

If $P ( h , k )$ be a point on the parabola $x=4 y^2$, which is nearest to the point $Q (0,33)$, then the distance of $P$ from the directrix of the parabola $y^2=4(x+y)$ is equal to :

Let a tangent to the curve $y^2=24 x$ meet the curve $x y=2$ at the points $A$ and $B$ Then the mid points of such line segments $A B$ lie on a parabola with the

Let R be the focus of the parabola y² = 20x and the line y=mx+c intersect the parabola at two points Pand Q. Let the point G(10,10) be the centroid of the triangle PQR. If c-m=6, then (PQ)² is

Let y = p(x) be the parabola passing through the points (–1, 0), (0, 1) and (1, 0). If the area of the region \(\{x,y):(x+1)^2+(y-1)^2≤1,y≤p(x)\}\) is A, then 12(π-4A) is equal to__.

If the line y = 4 + kx, k > 0, is the tangent to the parabola y = x – x² at the point P and V is the vertex of the parabola, then the slope of the line through P and V is:

Let the normal at the point P on the parabola y² = 6x pass through the point (5, –8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is

If the equation of the parabola, whose vertex is at \((5, 4)\) and the directrix is \(3x + y – 29 = 0\), is \(x^2 + ay^2 + bxy + cx + dy + k = 0\), then \(a + b + c + d + k\) is equal to

If vertex of a parabola is (2, –1) and the equation of its directrix is 4x – 3y = 21, then the length of its latus rectum is :

The sum of diameters of the circles that touch (i) the parabola 75x² = 64(5y – 3) at the point (\(\frac{8}{5},\frac{6}{5}\)) and (ii) the y-axis is equal to _______.

If the tangents drawn at the points \(P\) and \(Q\) on the parabola \(y^2=2 x-3\) intersect at the point \(R (0,1)\), then the orthocentre of the triangle \(PQR\) is :