Parabola

Let T be the tangent to the parabola $y^2 = 16x$ at the point $(64, 32)$. Let L be the tangent to the same parabola at another point $(x_1, y_1)$ on the parabola. If L and T are perpendicular to each other, then the distance between the poi...

Let $S$ denote the locus of the mid-points 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 quadra...

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 through $(0,-\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ inte...

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 $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statemen...

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 triangle $P F Q$, where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the nor...

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 the parabola at \(P_{1}\) and \(P_{2}\). Let \(Q_{1}\) and \(Q_{2}\) be points on the lin...

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. Let F be the focus of E. Then which of the following statements is(are) TRUE?

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 ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) passes through the poin...

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 \(-\) h)2 + (y \(-\) k)2 = r2 satisfies the following conditions :(i) Centre of C3 is collinear...

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 \(-\) h)2 + (y \(-\) k)2 = r2 satisfies the following conditions :(i) centre of C3 is collinear ...

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) possible value(s) of p, h and k?

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 only CORRECT combination?

Let \(P\) be the point on the parabola \({y^2} = 4x\) which is at the shortest distance from the center \(S\) of the circle \({x^2} + {y^2} - 4x - 16y + 64 = 0\). Let \(Q\) be the point on the circle dividing the line segment \(SP\) interna...

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 tangent to the circle \({C_1}\), at \(P\) touches other two circles \({C_2}\) and $${C_3}...

Suppose that the foci of the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 5} = 1\) are \(\left( {{f_1},0} \right)\) and \(\left( {{f_2},0} \right)\) where \({{f_1} > 0}\) and \({{f_2} < 0}\). Let \({P_1}\) and \({P_2}\) be two parabolas wit...

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 intersection of \(C\) with the line \(y=-5\), then the distance between \(A\) and \(B\) is

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 the parabola. If \(P\) lies in the first quadrant and the area of the triangle $$\Delta OPQ$...

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

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 \(S\,\,\left( {a{s^2},2as} \right)\) be distinct points on the parabola \({y^2} = 4ax\). Suppose that $$PQ...

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 \(S\,\,\left( {a{s^2},2as} \right)\) be distinct points on the parabola \({y^2} = 4ax\). Suppose that $$PQ...

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 \(y=2x+a\), \(a>0\).
If chord \(PQ\) subtends an angle \(\theta\) at the vertex of $${y^2} = 4a...

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 \(F\left( {{x_0},{y_0}} \right)\). The tangent to the parabola at \(F\left( {{x_0},{y_0}} \right)\) intersects t...

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 \(y=2x+a\), \(a>0\).
Length of chord \(PQ\) is

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 parabola. The area of the triangle \(PQS\) is

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 ratio \(1 : 3\). Then the locus of \(P\) is

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

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 \(P\left( {{1 \over 2},2} \right)\) on the parabola and \({\Delta _2}\) be the area of the tria...

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 \(AB\) as its diameter, then the slope of the line joining \(A\) and \(B\) can be

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 locus of the centroid of the triangle \(PTN\) is a parabola whose

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 \(y = 0\), where \(p \ne q\), is :

STATEMENT - 1 : The curve \(y=\frac{-x^{2}}{2}+x+1\) is symmetric with respect to the line \(x=1\).
STATEMENT - 2 : A parabola is symmetric about its axis.

The radius of the incircle of the triangle PQR is

The radius of the circumcircle of the triangle PRS is

The ratio of the areas of the triangles PQS and PQR is

The tangent to the curve \(y=e^x\) drawn at the point (\(c,e^c\)) intersects the line joining the points (\(c-1,e^{c-1}\)) and (\(c+1,e^{c+1}\))

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, respectively. Tangent to the circle at \(P\) and \(Q\) intersect the \(x\)-axis at \(R\) and...

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, respectively. Tangent to the circle at \(P\) and \(Q\) intersect the \(x\)-axis at \(R\) and...

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, respectively. Tangent to the circle at \(P\) and \(Q\) intersect the \(x\)-axis at \(R\) and...

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 symmetric about its axis.

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

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 \(2\sqrt 2\) respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola ...

\(\text { Normals are drawn at points } \mathrm{P}, \mathrm{Q} \text { and } \mathrm{R} \text { lying on the parabola } y^2=4 x \text { which intersect at }(3,0) \text {. Then }\)
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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 coordinates of \(Q\) are

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

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 it divides \(QP\) externally in the ratio \(1/2:1\). Find the locus of point \(R\)

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

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 \({m_1}\) \({m_2}\)\(= \alpha\) is a part of the parabola itself then find \(\alpha\).

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