Coordinate Geometry

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

If $a,b,c$ are the roots of the equation $x^3-3px^2+3qx-1=0$, then the centroid of the triangle with vertices $\left(a,\frac1a\right),\left(b,\frac1b\right),\left(c,\frac1c\right)$ is the point

A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts $p$ and $q$. Which of the following is true?

The straight lines $\dfrac{x}{a} + \dfrac{y}{b} = k$ and $\dfrac{x}{a} + \dfrac{y}{b} = \dfrac{1}{k}$ (with $k\neq0$) meet on:

$ABC$ is isosceles with $AB = AC$. $BC$ is parallel to x-axis. $m_1, m_2$ are slopes of the medians from $B$ and $C$. Then:

If the distance of $(x,y)$ from the origin is defined as $d(x,y) = \max(|x|,|y|)$, then the locus of points where $d(x,y)=1$ is:

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

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

Number of distinct solutions of $ x^{2} = y^{2} $ and $ (x - a)^{2} + y^{2} = 1 $ where $a$ is any real number:

Lines $2x + 3y - 6 = 0$ and $9x + 6y - 18 = 0$ cut coordinate axes in concyclic points. Center of circle is:

Area enclosed by $|x|+|y|=1$

Find the equation of the circle which passes through (–1, 1) and (2, 1), and having centre on the line x + 2y + 3 = 0 .

The equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, –1). The length of the side of the triangle is.

The lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangent to the same circle. The radius of the this circle is.

The equations of the line parallel to the line $2x – 3y = 7$ and passing through the middle point of the line segment joining the points (1, 3) and (1, –7) is.

The area enclosed within the curve $|X|+|Y| = 1$ (in square units) is

If the straight line ax + by + c = 0 always passes through (1, –2), then a, b, c are in

The median AD of ΔABC is bisected at E and BE is produced to meet the side AC at F. Then, AF ∶ FC is

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

An equilateral triangle is inscribed in the parabola $y^{2} = 4ax$, such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is:

If two circles $x^{2}+y^{2}+2gx+2fy=0$ and $x^{2}+y^{2}+2g'x+2f'y=0$ touch each other then whichof the following is true?

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

A circle touches the X-axis and also touches another circle with centre at (0, 3) and radius 2. Then the locus of the centre of the first circle is

If $(– 4, 5)$ is one vertex and $7x – y + 8 = 0$ is one diagonal of a square, then the equation of the other diagonal is

If (2, 1), (–1, –2), (3, 3) are the midpoints of the sides BC, CA, AB of a triangle ABC, then equation of the line BC is

The foot of the perpendicular from the point (2, 4) upon $x + y = 1$ 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

If the lines x + (a – 1)y + 1 = 0 and 2x + a²y – 1 = 0 are perpendicular, then the condition satisfies by a is

A line passing through (4, 2) meets the x and y-axis at P and Q respectively. If O is the origin, then the locus of the centre of the circumcircle of ΔOPQ is -

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

The equation of the circle passing through the point (4,6) and whose diameters are along x + 2y - 5 =0 and 3x - y - 1=0 is

The curve satisfying the differential equation ydx-(x+3y²)dy=0 and passing through the point (1,1) also passes through the point __________

The tangent at the point (2,  -2) to the curve $x^2 y^2-2x=4(1-y)$ does not passes through the point

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:

If P(1,2), Q(4,6), R(5,7) and S(a,b) are the vertices of a parallelogram PQRS, then

The lines $px+qy=1$ and $qx+py=1$ are respectively the sides AB, AC of the triangle ABC and the base BC is bisected at $(p,q)$. Equation of the median of the triangle through the vertex A is

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 $(-1,4)$.

If $(4,-3)$ and $(-9,7)$ are two vertices of a triangle and $(1,4)$ is its centroid, find the area of the triangle.

The equation of the plane passing through the point $(1,2,3)$ and having the normal vector $N = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ is:

The point on the curve $y = 6x - x^2$ where the tangent is parallel to the x-axis is:

Normal to the curve $y = x^3 - 3x + 2$ at the point $(2,4)$ is:

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 $x^2 + 2x – 4y^2 + 8y – 7 = 0$

The equation of the tangent at any point of curve $x=a cos2t, y=2\sqrt{2} a sint$ with $m$ as its slope is

Two friends A and B were standing at the diagonally opposite corners of a rectangular plot whose perimeter is 100m. A first walked x meters along the length of the plot towards East and then y meters towards the South. B walked x meters along the breadth towards North and then y meters towards West. At the end of their walks, A and B were standing at the diagonally opposite corners of a smaller rectangular plot whose perimeter is 40m. How much distance did A walk?

The range of values of θ in the interval (0, π) such that the points (3,5) and (sinθ, cosθ) lie on the same side of the line x + y − 1 = 0, is

$\theta={\cos }^{-1}\Bigg{(}\frac{3}{\sqrt[]{10}}\Bigg{)}$ is the angle between $\vec{a}=\hat{i}-2x\hat{j}+2y\hat{k}$ & $\vec{b}=x\hat{i}+\hat{j}+y\hat{k}$ then possible values of (x,y) that lie on the locus

Let a, b, c, d be no zero numbers. If the point of intersection of the line 4ax + 2ay + c = 0 & 5bx + 2by + d=0 lies in the fourth quadrant and is equidistance from the two are then

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?

If the line $a^2 x + ay +1=0$, for some real number $a$, is normal to the curve $xy=1$ then

Lines $L_1, L_2, .., L_10 $are distinct among which the lines $L_2, L_4, L_6, L_8, L_{10}$ are parallel to each other and the lines $L_1, L_3, L_5, L_7, L_9$ pass through a given point C. The number of point of intersection of pairs of lines from the complete set $L_1, L_2, L_3, ..., L_{10}$ is

Region R is defined as region in first quadrant satisfying the condition $x^2 + y^2 < 4$. Given that a point P=(r,s) lies in R, what is the probability that r>s?

The points (1,1/2) and (3,-1/2) are

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

For what values of $\lambda$ does the equation $6x^2 - xy + \lambda y^2 = 0$ represents two perpendicular lines and two lines inclined at an angle of $\pi/4$.

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

The obtuse angle between lines 2y = x + 1 and y = 3x + 2 is

The circle $x^2 + y^2+ \alpha x+ \beta y+ \gamma=0$ is the image of the circle $x^2 + y^2- 6x- 10y+ 30=0$ across the line 3x + y = 2. The value of $[\alpha+ \beta+ \gamma]$ is (where [.] represents the floor function.)

A circle with its center in the first quadrant touches both the coordinate axes and the line x-y-2=0. Then the area of the circle 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

Let the line $\frac{x}{4}+\frac{y}{2}=1$ meets the x-axis and y-axis at A and B, respectively. M is the midpoint of side AB, and M' is the image of the point M across the line x + y = 1. Let the point P lie on the line x + y = 1 such that the $\Delta$ABP is an isosceles triangle with AP = BP. Then the distance between M' and P is