Conic Sections Questions (164)

$(x-1)(y-2) = 5$ and $(x-1)^2 + (y+2)^2 = r^2$ intersect at four points $A, B, C, D$ and if centroid of $\triangle ABC$ lies on line $y = 3x - 4$, then locus of $D$ is:

Let $S$ be the focus of $y^2 = 4x$ and a point $P$ is moving on the curve such that its abscissa is increasing at the rate of 4 units/sec, then the rate of increase of projection of $SP$ on $x + y = 1$ when $P$ is at $(4, 4)$ is:

If two distinct tangents can be drawn from the point $(a, 2)$ on different branches of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, then

Area of the triangle formed by the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and any tangent to the hyperbola is $a^2\tan\lambda$ in magnitude then its eccentricity is:

$(x-1)(y-2) = 5$ and $(x-1)^2 + (y+2)^2 = r^2$ intersect at four points $A, B, C, D$ and if centroid of $\triangle ABC$ lies on line $y = 3x - 4$, then locus of $D$ is:

If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, $S_1$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1S_2S_3S_4$ is:

If origin is shifted to point $\left(3, \frac{7}{2}\right)$ and the axes are rotated through an angle $\theta$ in clockwise sense so that equation of given hyperbola changes to the standard form $\frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1$, then $\theta$ is:

Let $A\left(\frac{1}{2}, 0\right), B\left(\frac{3}{2}, 0\right), C\left(\frac{5}{2}, 0\right)$ be the given points and $P$ be a point satisfying $\max(PA + PB, PB + PC) < 2$. All points $P$ are points common to:

The extremities of latus rectum of a parabola are $(1, 1)$ and $(1, -1)$, then the equation of the parabola can be :

Parabola $y^2 = 4x$ and the circle having its centre at $(6, 5)$ intersect at right angle. Possible point of intersection of these curves can be :

A normal drawn to parabola $y^2 = 4ax$ meet the curve again at $Q$ such that angle subtended by $PQ$ at vertex is $90°$, then coordinates of $P$ can be :

If a pair of variable straight lines $x^2 + 4y^2 + \alpha xy = 0$ (where $\alpha$ is a real parameter) cut the ellipse $x^2 + 4y^2 = 4$ at two points A and B, then the locus of the point of intersection of tangents at A and B is:

Consider the ellipse $\frac{x^2}{t(k^2 + 2k + 5)} + \frac{y^2}{t(k+1)} = 1$ and $f(x)$ is a positive decreasing function, then:

If the tangent drawn at point $(t^2, 2t)$ on the parabola $y^2 = 4x$ is same as the normal drawn at point $(\sqrt{5}\cos\theta, 2\sin\theta)$ on the ellipse $4x^2 + 5y^2 = 20$. Then: