Conic Sections

Equation of the common tangent touching the circle $(x-3)^2+y^2=9$ and the parabola $y^2=4x$ above the $x$-axis is

The smaller area bounded by $y = 2 - x$ and $x^2 + y^2 = 4$ is:

Eccentricity of ellipse $ 9x^{2} + 5y^{2} - 30y = 0 $

Vertex of parabola $ y^{2} - 8y + 19 = 0 $

The sum of the focal distances of any point on the ellipse $\frac{x^{2}}{a^{2}} +\frac{y^{2}}{b^{2}} =1$ with eccentricity $e$ is given by

If PQ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ such that OPQ is an equilateral triangle, where O is the centre of the hyperbola, then which of the following is true?

The condition that the line lx + my + n = 0 becomes a tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ , is

A normal to the curve $x^{2} = 4y$ passes through the point (1, 2). The distance of the origin from the normal is

If the foci of the ellipse $b^{2}x^{2}+16y^{2}=16b^{2}$ and the hyperbola $81x^{2}-144y^{2}=\frac{81 \times 144}{25}$ coincide, then the value of $b$, is

If $x = 1$ is the directrix of the parabola $y^{2} = kx - 8$, then k is:

The locus of the intersection of the two lines $\sqrt{3} x-y=4k\sqrt{3}$ and $k(\sqrt{3}x+y)=4\sqrt{3}$, for different values of k, is a hyperbola. The eccentricity of the hyperbola is:

The radius of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}$and having it centre at (0, 3) is

If $3x + 4y + k = 0$ is a tangent to the hyperbola ,$9x^{2}-16y^{2}=144$ then the value of $K$ is

The foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ and the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{{81}}=\frac{1}{25}$ coincide, then the value of $b^{2}$ is

The locus of the mid points of all chords of the parabola $y^{2}=4x$ which are drawn through its vertex, is

Equation of the tangent from the point (3,−1) to the ellipse 2x² + 9y² = 3 is

If S and S' are foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, B is the end of the minor axis and BSS' is an equilateral triangle, then the eccentricity of the ellipse is

Find the number of point(s) of intersection of the ellipse $\dfrac{x^2}{4}+\dfrac{(y-1)^2}{9}=1$ and the circle  x² + y² = 4

The tangent to an ellipse x² + 16y² = 16 and making angel 60° with X-axis is:

The eccentric angle of the extremities of latus-rectum of the ellipse $\frac{{x}^2}{{a}^2}^{}+\frac{{y}^2}{{b}^2}^{}=1$ are given by

The locus of the point of intersection of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which meet right angles is

The number of common tangents to the circle  $x^2+y^2=4$ and $x^2+y^2-6x-8y=24$ is

Focus of the parabola $x^2 + y^2 - 2xy - 4(x + y - 1) = 0$ is:

If the circles $ x^2 + y^2 + 2x + 2ky + 6 = 0$ and $x^2 + y^2 + 2ky + k = 0$ intersect orthogonally, then $k$ is:

The equation of ellipse with major axis along the x–axis and passes through the point $(4,3)$ and $(-1,4)$.

The locus of the mid-point of all chords of the parabola $y^2 = 4x$ which are drawn through its vertex is

Find foci of the equation $x^2 + 2x – 4y^2 + 8y – 7 = 0$

A point P in the first quadrant, lies on $y^2 = 4ax$, a > 0, and keeps a distance of 5a units from its focus. Which of the following points lies on the locus of P?

The equation $3x^2 + 10xy + 11y^2 + 14x + 12y + 5 = 0$ represents

The two parabolas $y^2 = 4a(x + c)$ and $y^2 = 4bx, a > b > 0$ cannot have a common normal unless

If (4, 3) and (12, 5) are the two foci of an ellipse passing through the origin, then the eccentricity of the ellipse is

Let $F_1, F_2$ be foci of hyperbola $\frac{{x}^2}{{a}^2}-\frac{{y}^2}{{b}^2}=1$, a>0, b>0, and let O be the origin. Let M be an arbitrary point on curve C and above X-axis and H be a point on $MF_1$ such that $MF_2\perp{{F}}_1{{F}}_2$, $MF_1\perp{{O}}{{H}}$, $|OH|=\lambda |OF_2|$ with $\lambda \in(2/5, 3/5)$, then the range of the eccentricity $e$ is

An equilateral triangle is inscribed in the parabola $y^2 = x$. One vertex of the triangle is at the vertex of the parabola. The centroid of triangle is