Parabola · Mathematics · JEE Advanced

Let $A_1, B_1, C_1$ be three points in the $x y$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8 x$ at $A_1$ an...

Consider the parabola $$y^{2}=4 x$$. Let $$S$$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $$P=(-2,1)$$ meet...

Let E denote the parabola y2 = 8x. Let P = ($$-$$2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E....

If a chord, which is not a tangent, of the parabola y2 = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) po...

Let $$P$$ be the point on the parabola $${y^2} = 4x$$ which is at the shortest distance from the center $$S$$ of the circle $${x^2} + {y^2} - 4x - 16y...

The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ in the first quadrant, Let the...

Let $$P$$ and $$Q$$ be distinct points on the parabola $${y^2} = 2x$$ such that a circle with $$PQ$$ as diameter passes through the vertex $$O$$ of th...

Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by

Let $$A$$ and $$B$$ be two distinct points on the parabola $${y^2} = 4x$$. If the axis of the parabola touches a circle of radius $$r$$ having $$AB$$ ...

The tangent $$PT$$ and the normal $$PN$$ to the parabola $${y^2} = 4ax$$ at a point $$P$$ on it meet its axis at points $$T$$ and $$N$$, respectively....

The equations of the common tangents to the parabola $$y = {x^2}$$ and $$y = - {\left( {x - 2} \right)^2}$$ is/are

A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha)$ to the parabola $x^2=-4 a y$, where $a>0$. Let $L$ be the line passing ...

Suppose that the foci of the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 5} = 1$$ are $$\left( {{f_1},0} \right)$$ and $$\left( {{f_2},0} \right)$$ w...

If the normals of the parabola $${y^2} = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $${\left( {x - 3} \right)^2} + {\...

Let the curve $$C$$ be the mirror image of the parabola $${y^2} = 4x$$ with respect to the line $$x+y+4=0$$. If $$A$$ and $$B$$ are the points of inte...

Let $$S$$ be the focus of the parabola $${y^2} = 8x$$ and let $$PQ$$ be the common chord of the circle $${x^2} + {y^2} - 2x - 4y = 0$$ and the given p...

Consider the parabola $${y^2} = 8x$$. Let $${\Delta _1}$$ be the area of the triangle formed by the end points of its latus rectum and the point $$P\l...

Let $P$ be a point on the parabola $y^2=4 a x$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the tria...

Let a, b and $$\lambda $$ be positive real numbers. Suppose P is an end point of the latus return of the parabola y2 = 4$$\lambda $$x, and suppose the...

Let the circles C1 : x2 + y2 = 9 and C2 : (x $$-$$ 3)2 + (y $$-$$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $$-$$...

Let the circle C1 : x2 + y2 = 9 and C2 : (x $$-$$ 3)2 + (y $$-$$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $$-$$ ...

If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the o...

Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a...

Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a...

A line $$L:y=mx+3$$ meets $$y$$-axis at R$$(0, 3)$$ and the arc of the parabola $${y^2} = 16x,$$ $$0 \le y \le 6$$ at the point $$F\left( {{x_0},{y_0}...

Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+...

Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+...

Let $$(x, y)$$ be any point on the parabola $${y^2} = 4x$$. Let $$P$$ be the point that divides the line segment from $$(0, 0)$$ to $$(x, y)$$ in the ...

The locus of the orthocentre of the triangle formed by the lines $$(1 + p)x - py + p(1 + p) = 0, $$ $$(1 + q)x - qy + q(1 + q) = 0$$ and $$y = 0$$, w...

STATEMENT-1: The curve $$y = {{ - {x^2}} \over 2} + x + 1$$ is symmetric with respect to the line $$x=1$$. because STATEMENT-2: A parabola is symmetri...

Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, re...

Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, re...

Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, re...

The axis of a parabola is along the line $$y = x$$ and the distances of its vertex and focus from origin are $$\sqrt 2 $$ and $$2\sqrt 2 $$ respective...

Match the following : $$(3, 0)$$ is the pt. from which three normals are drawn to the parabola $${y^2} = 4x$$ which meet the parabola in the points $$...

Tangent to the curve $$y = {x^2} + 6$$ at a point $$(1, 7)$$ touches the circle $${x^2} + {y^2} + 16x + 12y + c = 0$$ at a point $$Q$$. Then the coord...

The angle between the tangents drawn from the point $$(1, 4)$$ to the parabola $${y^2} = 4x$$ is

The focal chord to $${y^2} = 16x$$ is tangent to $${\left( {x - 6} \right)^2} + {y^2} = 2,$$ then the possible values of the slope of the chord, are

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix

The equation of the common tangent to the curves $${y^2} = 8x$$ and $$xy = - 1$$ is

The equation of the directrix of the parabola $${y^2} + 4y + 4x + 2 = 0$$

The equation of the common tangent touching the circle $${\left( {x - 3} \right)^2} + {y^2} = 9$$ and the parabola $${y^2} = 4x$$ above the $$x$$-axis...

If the line $$x - 1 = 0$$ is the directrix of the parabola $${y^2} - kx + 8 = 0,$$ then one of the values of $$k$$ is

If $$x + y = k$$ is normal to $${y^2} = 12x,$$ then $$k$$ is

The curve described parametrically by $$x = {t^2} + t + 1,$$ $$y = {t^2} - t + 1 $$ represents

Consider a circle with its centre lying on the focus of the parabola $${y^2} = 2px$$ such that it touches the directrix of the parabola. Then a point ...

Tangent is drawn to parabola $${y^2} - 2y - 4x + 5 = 0$$ at a point $$P$$ which cuts the directrix at the point $$Q$$. $$A$$ point $$R$$ is such that ...

Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $$...

Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any point on $${C_1}$$ and $$Q$$ be a...

From a point $$A$$ common tangents are drawn to the circle $${x^2} + {y^2} = {a^2}/2$$ and parabola $${y^2} = 4ax$$. Find the area of the quadrilater...

Points $$A, B$$ and $$C$$ lie on the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$A, B$$ and $$C$$, taken in pairs, intersect at points...

Show that the locus of a point that divides a chord of slope $$2$$ of the parabola $${y^2} = 4x$$ internally in the ratio $$1:2$$ is a parabola. Find ...

Through the vertex $$O$$ of parabola $${y^2} = 4x$$, chords $$OP$$ and $$OQ$$ are drawn at right angles to one another . Show that for all positions o...

Three normals are drawn from the point $$(c, 0)$$ to the curve $${y^2} = x.$$ Show that $$c$$ must be greater than $$1/2$$. One normal is always the $...

$$A$$ is point on the parabola $${y^2} = 4ax$$. The normal at $$A$$ cuts the parabola again at point $$B$$. If $$AB$$ subtends a right angle at the ve...

Suppose that the normals drawn at three different points on the parabola $${y^2} = 4x$$ pass through the point $$(h, k)$$. Show that $$h>2$$.

The point of intersection of the tangents at the ends of the latus rectum of the parabola $${y^2} = 4x$$ is ...... .