List of top Mathematics Questions on Coordinate Geometry

If \(A, B, C\) are vertices of a triangle with position vectors \(\vec{a}, \vec{b}, \vec{c}\) respectively, then find the position vector of the point \(D\) where the angle bisector from vertex \(A\) meets \(BC\).

Evaluate: \(\tan^{-1}(1) + \tan^{-1}(4) + \tan^{-1}(5) + \tan^{-1}\left(\frac{1}{4}\right) = \pi + \tan^{-1}\left(\frac{\alpha}{2}\right)\). Find the value of \(\alpha\).

If \(y = f\left(\frac{3 + 2x}{3 - 2x}\right)\), where \(f(x) = \tan(\log x)\), and \(\frac{dy}{dx} = \frac{A}{B + Cx^2} \cdot \sec^2\left(\log \frac{3 + 2x}{3 - 2x}\right)\), then find \(A, B, C\).

The radical axis of the circles \[ x^2+y^2+4x+6y+7=0 \] and \[ 4x^2+4y^2+8x+12y-24=0 \] is a tangent to the circle \[ x^2+y^2=13 \] at a point \((\alpha,\beta)\), then \(\alpha+\beta=\)

If the locus of midpoints of the chords of the circle \[ S\equiv x^2+y^2-6x-8y-11=0 \] which subtend a right angle at \(A(1,2)\) is another circle \(S'=0\), then the centre of \(S'=0\)

Let \[ S\equiv x^2+y^2-6x+4y+c=0,\qquad S'\equiv x^2+y^2-4x+6y+9=0, \] \[ S''\equiv x^2+y^2+5x+3y+k=0 \] be three circles. If the angles of intersection of the circle \(S'=0\) with the circles \(S=0\) and \(S''=0\) are respectively \[ \frac{\pi}{4} \quad\text{and}\quad \frac{\pi}{2}, \] then \(c+k=\)

Let \(L=0\) be the chord of contact of \((5,1)\) with respect to the circle \[ S\equiv x^2+y^2+8x+10y-8=0. \] If the pole of \(L=0\) with respect to the circle \[ S'\equiv x^2+y^2+4y-21=0 \] is \((-k,-h)\), then \(k+h=\)

Equation of the image of the pair of lines \[ ax^2+2hxy+by^2=0 \] with respect to \(y=0\) is

The locus of the centre of a circle touching the lines \[ x+2y=0 \] and \[ x-2y=0 \] is

The line common to the two families of concurrent lines \[ x+y+\lambda(4x+3y-1)=0 \] and \[ 2x-3y+k(x+y-5)=0 \] is along the base of a triangle. If \((1,1)\) is the vertex of the triangle, then the perpendicular distance from that vertex to its base is

If the equation of the line joining the points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) is \[ ax+by=c \] and the distance between \(A\) and \(B\) is \(\sqrt{a^2+b^2}\), then \(c^2=\)

Two diagonal sides of a parallelogram \(ABCD\) are \[ x-y+1=0 \] and \[ 3x-y=9. \] If one of its diagonals is \[ 7x-5y+3=0, \] then the equation of the other diagonal is

The locus of midpoints of the chords of the circle \[ x^2+y^2-2x-2y+1=0 \] which are parallel to the line \[ x+y+2=0 \] is

The transformed equation of \[ 4x^2-4xy+7y^2-24=0 \] in the new coordinate system when the axes are rotated through an angle \(\theta\) about the origin in the positive direction is \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] If \[ 0<\theta<\frac{\pi}{4}, \] then \[ 1+b^2\sin^2\theta= \]

Given the linear equation \(y = 2x + k\), determine the value of k if the point P(3, 7) lies on the line.

Relation between x and y such that the point (x, y) is equidistant from the points (-7, -1) and (3, 5) is :

Point (0, -7) lies :

If R (5, 6) is the midpoint of the line segment AB joining the points A (6, 5) and B (4, y), then y equals to

The point P(-4, 6) divides the line segment joining A(-6, 10) and B(3, -8) internally in the ratio

The graph of the linear equation \(2x + 3y = 6\) is a line which meets the y-axis at the point :

The distance of the point \( A(4a, 3a) \) from x-axis is :

The mid-point of the line segment joining the points $(5, -4)$ and $(6, 4)$ lies on :

If \(A,B\) are the feet of the perpendiculars drawn from \(P(3,1)\) on the pair of lines \[ 2x^2+3xy-2y^2+3x+y+1=0 \] and \(Q\) is the point of intersection of the pair of lines, then the area of the quadrilateral \(PAQB\) is

Find the coordinates of a point on the line x + y = 5 which is equidistant from (6, 4) and (5, 2).

Diagonals AC and BD of square ABCD intersect at P. Coordinates of points B and D are (9, -2) and (1, 6) respectively. (i) Find the coordinates of point P. (ii) Find the length of the side of the square.