Coordinate Geometry Questions (181)

TP and TQ are tangents to a parabola and $p_1$, $p_2$, $p_3$ are the lengths of perpendiculars from $P$, $T$, $Q$ respectively on any tangent to the parabola. Then $p_1$, $p_2$, $p_3$ are in

Three circles of radii $a,b,c$ ($a<b<c$) touch each other externally. If they have $x$-axis as a common tangent, then

The coordinate of point $P$ on the line $3x+2y+10=0$ such that $|PA-PB|$ is maximum, where $A=(4,2)$ and $B=(2,4)$, is

A line $y = m(x-4)$ meets the $x$-axis at $P$ and the parabola $x^2 = 32y$ at $Q(x_1,y_1)$. The tangent to the parabola at $Q$ meets the $x$-axis at $R(x_2,0)$, $0 < x_2 < 6$. If the area of $\triangle PQR$ assumes a local maximum, then the value of $m$ is

In a parabola $y^2=4ax$, two points $P$ and $Q$ are taken such that the tangents drawn to the parabola at these points meet at the directrix at $R$. The focus (locus of circumcenter of $\triangle PQR$) will be

Consider three curves $H:(x+a)y=\lambda$, $\lambda<0$, $a>0$; $C: x^2+y^2-21y+109=0$; $P: y^2=bx$, $b>0$. Let the line $2x-y+8=0$ touch $H$, $C$ and $P$ at $L$, $M$, $N$ respectively such that $LM=MN=\sqrt{45}$. Then $8\lambda+b+a$ is

Let an ellipse have foci $S_1$ and $S_2$ with eccentricity $e = \frac{1}{2}$. For a point $P$ on the ellipse, let $\angle PS_1S_2 = \alpha$, $\angle PS_2S_1 = \beta$, $\angle S_1PS_2 = \gamma$. If $\cot\frac{\alpha}{2}$, $\cot\frac{\gamma}{2}$, $\cot\frac{\beta}{2}$ are in A.P., then $\cos(\alpha - \beta)$ is

Let $H_L$ and $H_R$ be two branches of a hyperbola with $A,B$ as endpoints of latus rectum on $H_L$. If $C$ is on $H_R$ such that $\angle ACB$ is never obtuse, then maximum possible eccentricity is

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

The equation of a circle passing through origin, making equal intercepts on coordinate axes and passing through $(4,3)$ is

Three circles of radii $a,b,c$ ($a<b<c$) touch each other externally. If they have $x$-axis as a common tangent, then

In $\triangle ABC$, $AB=AC$ where $A=(3,1)$ and the equation of the base $BC$ is $2x+y=4$. Also $B$ lies on $x+3y=7$. Then sum of coordinates of vertex $C$ is

If the circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius 5, in such a manner that the common chord is of maximum length and has a slope equal to $3/4$, then one of the co-ordinates of the centre of $C_2$ are

The number of integral values of $k$ for which the line $3x+4y-k=0$ lies between the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-12y+113=0$, without cutting a chord on either circle, is equal to

The equation of the circle having centre on the line $x+2y-3=0$ and passing through the points of intersection of circles $x^2+y^2-2x-4y+1=0$ and $x^2+y^2-4x-2y+4=0$

Find the largest value of $y/x$ for a point $(x,y)$ on the circle $(x-3)^2+(y-3)^2=6$.

Area bounded by the set of points $S=\{(x,y):\,||x|-1|+||y|-1|\leq1\}$ is

Let the foci of the Ellipse $\dfrac{x^2}{16}+\dfrac{y^2}{7}=1$ and Hyperbola $\dfrac{x^2}{144}-\dfrac{y^2}{\alpha}=\dfrac{1}{25}$ coincide. Then the length of the latus rectum of the Hyperbola is

In an ellipse with major axis along x-axis, distance between foci is 6 and minor axis has length 8. The eccentricity is

The radical centre of circles described on the three sides $4x-7y+10=0$, $x+y=5$, and $7x+4y=15$ of a triangle as diameters is

Let $x^2+y^2+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $A+C$ is equal to

If area bounded by an ellipse with its auxiliary circle equals area bounded by its auxiliary circle and director circle, then eccentricity of ellipse is

Let $(2,3)$ be the focus of a parabola and $x+y=0$ and $x-y=0$ be its two tangents, then equation of its directrix will be

Which is/are true: I) A variable circle cuts two fixed perpendicular lines so that each intercept is of different length, then locus of centre is hyperbola. II) For hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with focal chord at eccentric angles $\theta_1,\theta_2$: $e=\frac{\cos((\theta_1+\theta_2)/2)}{\cos((\theta_1-\theta_2)/2)}=\frac{\sin(\theta_1+\theta_2)}{\sin\theta_1+\sin\theta_2}$.

If the tangent at point $P$ on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets auxiliary circle at $Q$ and $R$, and $QR$ subtends right angle at centre, then eccentricity is

Find the locus of a point whose distance from $x$-axis is twice the distance from the point $(1,-1,2)$

Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7},3\right)$ intersect the circle at only one point $p(\alpha,\beta)$. Then $\dfrac{29}{6}(\beta-\alpha)$ is equal to

Number of normals to the ellipse $\dfrac{x^2}{4}+\dfrac{y^2}{1}=1$ passing through $(0,0)$ is

A rectangular hyperbola and ellipse have coincident endpoints of latus rectum. Eccentricity of ellipse equals

If the pair of perpendicular lines $4x^2 + by^2 + 2\cos\theta\cdot xy + 12x + 2\sin^2\theta\cdot y + c = 0$, $\theta \in \left(\frac{3\pi}{2}, 2\pi\right)$ intersect on the $x$-axis, then the area of the triangle formed by the given pair of lines and $y = 2$ is

The co-ordinate of the focus of the parabola described parametrically by $x=5t^2+2$, $y=10t+4$ is