Coordinate Geometry PYQ — Page 17

A ray of light through $(2,1)$ is reflected at a point $P$ on the $y-$ axis and then passes through the point $(5,3).$ If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}},$ then the equation of the other directrix can be:

A hyperbola passes through the foci of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

The set of all possible values of $\theta$ in the interval $(0,\pi )$ for which the points$(1,2)$ and $(\mathrm{sin}\theta ,\mathrm{cos}\theta )$ lie on the same side of the line $x+y=1$ is?

The numbers of integral values of $k$ for which the line, $3x+4y=k$ intersects the circle, ${x}^{2}+{y}^{2}-2x-4y+4=0$ at two distinct points is....

The locus of the mid-points of the perpendiculars drawn from points on the line $x=2y$, to the line $x=y$, is.

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola ${x}^{2}=4y$ internally in the ratio $1:2$ is:

The diameter of the circle, whose Centre lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$ is

The area (in sq. units) of an equilateral triangle inscribed in the parabola ${y}^{2}=8x,$ with one of its vertices on the vertex of this parabola is

Let two points be $A(1,-1)$ and $B(0,2).$ If a point $P(x\text{'},y\text{'})$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line $3x+y-4\lambda =0,$ then a value of $\lambda$ is

Let the tangents drawn from the origin to the circle, ${x}^{2}+{y}^{2}-8x-4y+16=0$ touch it at the points $A$ and $B$ . Then ${(AB)}^{2}$ is equal to

Let the latus rectum of the parabola ${y}^{2}=4x$ be the common chord to the circles ${C}_{1}$ and ${C}_{2}$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles ${C}_{1}$ and ${C}_{2}$ is :

Let $L$ denote the line in the $xy$-plane with $x$ and $y$ intercepts as $3$ and $1$ respectively. Then the image of the point$(-1,-4)$ in the line is :

Let $PQ$ be a diameter of the circle ${x}^{2}+{y}^{2}=9$. If $\alpha$ and $\beta$ are the lengths of the perpendiculars from $P$ and $Q$ on the straight line, $x+y=2$ respectively, then the maximum value of $\alpha \beta$ is _______

Let $A(1,0),B(6,2)$ and $C(\frac{3}{2},6)$ be the vertices of a triangle $ABC.$ If $P$ is a point inside the triangle $ABC$ such that the triangles $APC,APB$ and $BPC$ have equal areas, then the length of the line segment $PQ,$ where $Q$ is the point $(-\frac{7}{6},-\frac{1}{3}),$ is

Let ${e}_{1}$ and ${e}_{2}$ be the eccentricities of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{{b}^{2}}=1(b<5)$ and the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{{b}^{2}}=1$ respectively satisfying ${e}_{1}{e}_{2}=1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(\alpha ,\beta )$ is equal to:

The locus of the centre of a circle which touches the line x=1 and passes through the origin is:

If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k,3)$ has $y$-intercept equal to $-4$, then a value of $k$ is;

If the line, $2x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4x-2y+\alpha =0$ and $6x-3y+\beta =0$ respectively, then the sum of all possible values of $\alpha$ and $\beta$ is ____________.

If the length of the chord of the circle, ${x}^{2}+{y}^{2}={r}^{2}(r>0)$ along the line, $y-2x=3$ is $r$, then ${r}^{2}$ is equal to:

If the distance between the foci of an ellipse is $6$ and the distance between its directrix is $12,$ then the length of its latus rectum is

If the co-ordinates of two points $A$ and $B$ are $(\sqrt{7},0)$ and $(-\sqrt{7},0)$ respectively and $P$ is any point on the conic, $9{x}^{2}+16{y}^{2}=144,$ then $PA+PB$ is equal to :

If one end of a focal chord $AB$ of the parabola ${y}^{2}=8x$ is at $A(\frac{1}{2},-2),$ then the equation of the tangent to it at $B$ is:

If ${e}_{1}$ and ${e}_{2}$ are the eccentricities of the ellipse $\frac{{x}^{2}}{18}+\frac{{y}^{2}}{4}=1$ and the hyperbola $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{4}=1$ respectively and $({e}_{1},{e}_{2})$ is a point on the ellipse $15{x}^{2}+3{y}^{2}=k$ , then the value of $k$ is equal to

If a line $y=mx+c$, is a tangent to the circle ${(x-3)}^{2}+{y}^{2}=1$, and it is perpendicular to a line ${L}_{1},$ where ${L}_{1}$ is the tangent to the circle ${x}^{2}+{y}^{2}=1$, at the point $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$, then