List of top Mathematics Questions on Parabola

An equilateral triangle OAB is inscribed in the parabola $y^2 = 4x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

Let P be the parabola, whose focus is $ (-2, 1) $ and directrix is $ 2x + y + 2 = 0 $. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:

Two parabolas have the same focus $(4, 3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:

Let $ y^2 = 12x $ be the parabola and $ S $ its focus. Let $ PQ $ be a focal chord of the parabola such that $ (SP)(SQ) = \frac{147}{4} $. Let $ C $ be the circle described by taking $ PQ $ as a diameter. If the equation of the circle $ C $ is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then $ \beta - \alpha $ is equal to:

Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^2 = 16x $ be $ (1, -4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m : n $, gcd($m, n$) = 1, then $ m^2 + n^2 $ is equal to:

Let the focal chord PQ of the parabola $ y^2 = 4x $ make an angle of $ 60^\circ $ with the positive x-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, $ S $ being the focus of the parabola, touches the y-axis at the point $ (0, \alpha) $, then $ 5\alpha^2 $ is equal to:

Let $ P(4, 4\sqrt{3}) $ be a point on the parabola $ y^2 = 4ax $ and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

If the line $ 3x - 2y + 12 = 0 $ intersects the parabola $ 4y = 3x^2 $ at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to:

Let A and B be the two points of intersection of the line $ y + 5 = 0 $ and the mirror image of the parabola $ y^2 = 4x $ with respect to the line $ x + y + 4 = 0 $. If $ d $ denotes the distance between A and B, and $ a $ denotes the area of $ \Delta SAB $, where $ S $ is the focus of the parabola $ y^2 = 4x $, then the value of $ (a + d) $ is:

Let $ P(4, 4\sqrt{3}) $ be a point on the parabola $ y^2 = 4ax $ and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

If the shortest distance of the parabola $y^{2}=4x$ from the centre of the circle $x² + y² - 4x - 16y + 64 = 0$ is d, then d²is equal to:

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

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 $ 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 $?

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 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 _____

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

Find the area of the region bounded by the parabola $$ y^2 = 4ax \text{ and its latus rectum.} $$

Let $L_1$ be the length of the common chord of the curves \[ x^2 + y^2 = 9 \quad {and} \quad y^2 = 8x \] and let $L_2$ be the length of the latus rectum of $y^2 = 8x$. Then:

If the focus of the parabola \[ (y - k)^2 = 4(x - h) \] always lies between the lines $x + y = 1$ and $x + y = 3$ then:

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