List of top Mathematics Questions on Coordinate Geometry asked in TS EAMCET

The straight line joining the points \[ (2,-2,1) \quad\text{and}\quad (-2,1,-1) \] is perpendicular to the plane \[ \pi_1 \] which passes through the point \[ (1,1,1) \] and the equation of \(\pi_1\) is \[ ax+by+cz+d=0. \] If \[ \pi_2 \] is the plane passing through the point \[ (1,2,3) \] whose normal vector is \[ (a,b,c), \] then the equation of \(\pi_2\) is

Let \[ A=(2,0,3),\qquad B=(0,1,4),\qquad C=(5,6,0) \] be three points. If \(L_1\) and \(L_2\) are the lines bisecting the angles between \(AB\) and \(AC\), then the direction ratios of the line perpendicular to both \(L_1\) and \(L_2\) are

If the points \[ (\alpha,4,2),\ (6,\beta,-1)\ \text{and}\ (8,\beta-7) \] are collinear, then \(\alpha+\beta=\)

The length of the common chord of the circles \[ x^2+y^2-8x-6y-11=0 \] and \[ x^2+y^2-6x+8y+9=0 \] is

Let a tangent \(L_1\) with slope \(m\) drawn to the parabola \[ y^2=8x \] be perpendicular to a normal \(L_2\) drawn to the parabola \[ y^2=12x. \] If \(m=1\) and the point of intersection of \(L_1\) and \(L_2\) is \((\alpha,\beta)\), then \(\alpha+\beta=\)

Let \(\dfrac32\) be the eccentricity of the conjugate hyperbola of a hyperbola \(H\) and one of the foci of \(H\) lie on the straight line \[ x+y-3=0. \] If the transverse and conjugate axes of \(H\) are along \(X\)-axis and \(Y\)-axis respectively, then the length of the latus rectum of \(H\) is

If the length of a focal chord drawn to the parabola \[ y^2=64x \] is \(289\) and the angle made by this focal chord with the positive \(X\)-axis measured in the positive direction is an acute angle, then the slope of the focal chord is

If the major axis of an ellipse is parallel to the \(X\)-axis and the ratio of the lengths of its latus rectum and minor axis is \(2:3\), then the ratio of distances from its centre to its focus and directrix is

For the ellipse \[ \frac{x^2}{36}+\frac{y^2}{25}=1, \] the straight line \[ 2x+y-5=0 \] is

If the pair of lines joining the origin to the points of intersection of the curve \[ 2x^2-5xy-3y^2-10x+27y-8=0 \] and the straight line \[ ax+y=1 \] are perpendicular to each other and \(a\) is not an integer, then \(a=\)

The parametric equation of a circle passing through origin are given by \[ x=-g+5\cos\theta, \qquad y=-f+5\sin\theta. \] If the straight line passing through origin with slope \[ -\frac43 \] is the diameter of this circle, then the sum of the intercepts made by this circle on the coordinate axes is

If the length of the chord of the line \[ x+y-1=0 \] of the circle \[ x^2+y^2-6x+2fy-2=0\qquad(f>0) \] is \[ \frac{6\sqrt7}{\sqrt2}, \] then the length of the intercept made by this circle on the \(Y\)-axis is

If the straight line \[ x+by+1=0 \] is a polar with respect to the circle \[ x^2+y^2-8x+10y-8=0, \] but not a tangent and not a chord, then all the values of \(b\) lie in the interval

If the equation of a transverse common tangent drawn to the circles \[ x^2+y^2-2x-10y+1=0 \] and \[ x^2+y^2+8x+14y+1=0 \] is \[ 5x+by+c=0, \] then \(b+c=\)

If the circles \[ x^2+y^2-10x+8y+5=0 \] and \[ x^2+y^2+6x-4y+c=0 \] cut each other orthogonally, then the sum of the radii of these circles is

If the distance of a variable point \(P\) from a fixed point \((3,-4)\) is \(\dfrac23\) times the distance of \(P\) from the fixed line \(x-y+2=0\), then the locus of \(P\) is \[ ax^2+4by+by^2-62x+80y+c=0, \] then \(2c=\)

When axes are rotated through an angle \(\theta\) about the origin in the positive direction, if the equation \[ 3x^2+\sqrt{3}xy-5=0 \] is transformed to the form \[ ax'^2+by'^2=10, \] then \(ab=\)

Two straight lines are drawn parallel to the straight line \[ L\equiv 5x-12y-13=0 \] which are at a distance of \(13\) units from the given line. Among these lines, if \[ ax+by+c=0 \] is the line which is closest to \[ L=0, \] then \[ \frac{a-2b+c}{a+b} = \]

If \[ x+y-2=0 \] and \[ 2x-y-1=0 \] represent two adjacent sides of a parallelogram and \[ x+4y-14=0 \] represents one of its diagonals, then one of the vertices of the parallelogram is

If the point \(M\) is the foot of the perpendicular drawn from the point \(P(0,9)\) to the straight line \[ 2x-5y+16=0, \] and \(Q=(12,8)\), then the orthocentre of \(\triangle MPQ\) is

If the sum of the squares of two consecutive positive odd integers is 1354, then one of them is:

If the latus rectum through one of the foci of a hyperbola \( \frac{x^2}{9} - \frac{y^2}{b^2} = 1 \) subtends a right angle at the farther vertex of the hyperbola, then \(b^2 = \)

Let e be the eccentricity of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). If a=5, b=4 and the equation of the normal drawn at one end of the latus rectum that lies in the first quadrant is \(lx+my=27\), then l+m=

If P is any point on the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) and S, S' are its foci, then the maximum area (in sq. units) of \(\Delta SPS' = \)

A normal chord PQ drawn at a point P on the parabola \(y^2=5x\) subtends a right angle at the vertex. If P lies in the first quadrant, then the other end Q of the normal chord is