Coordinate Geometry Questions

Let $A(h,k)$, $B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with $AC$ as its hypotenuse. If the area of the triangle is $1$, then the set of values which $k$ can take is given by:

A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is:

The two lines $x=ay+b,z=cy+d$; and $x=a^{\prime }y+b^{\prime },z=c^{\prime }y+d^{\prime }$ are perpendicular to each other if:

The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is:

If $a\ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay,$ then

In an ellipse, the distance between its foci is $6$ and the minor axis is $8$. Then its eccentricity is:

The intercept on the line $y=x$ by the circle $x^2+y^2-2x=0$ is $AB.$ Equation of the circle on $AB$ as a diameter is

If the lines $2x+3y+1=0$ and $3x-y-4=0$ lie along diameters of a circle of circumference $10\pi ,$ then the equation of the circle is

Angle between the tangents to the curve $y=x^2-5x+6$ at the points $(2,0)$ and $(3,0)$ is:

If the lines $3x-4y-7=0$ and $2x-3y-5=0$ are two diameters of a circle of area $49\pi$ square units, the equation of the circle is:

The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}.$ If one of the directrices is $x=4,$ then the equation of the ellipse is

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $2\pi /3$ at its centre is:

The image of the point $(-1,3,4)$ in the plane $x-2y=0$ is:

Consider a family of circles which are passing through the point $(-1,1)$ and are tangent to the $x$ -axis. If $(h,k)$ are the coordinates of the centre of the circles, then the set of values of $k$ is given by the interval:

The normal to a curve at $P(x,y)$ meets the $x$ -axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a:

Let $P=(-1,0)$, $Q=(0,0)$ and $R=(3,3\sqrt{3})$ be three points. The equation of the bisector of the angle $PQR$ is:

If one of the lines of $m{y}^2+(1-m^2)xy-m{x}^2=0$ is a bisector of the angle between the lines $xy=0$, then $m$ is:

Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8x$. The locus of midpoint of $PQ$ is:

The point diametrically opposite to the point $P(1,0)$ on the circle $x^2+y^2+2x+4y-3=0$ is:

The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k,3)$ has $y$ -intercept $-4$. Then a possible value of $k$ is:

The lines $(p+p^2)x-y+q=0$ and $(p^2+1)x+(p^2+1)y+2q=0$ are perpendicular to a common line for:

Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3y-\alpha z+\beta =0$. Then $(\alpha ,\beta )$ equals:

If $P$ and $Q$ are the points of intersection of the circles $x^2+y^2+3x+7y+2p-5=0$ and $x^2+y^2+2x+2y-p^2=0$, then there is a circle passing through $P,Q$ and $(1,1)$ for:

Three distinct points $A,B$ and $C$ are given such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $1/3$. Then the circumcentre of the triangle $ABC$ is at the point:

The ellipse $x^2+4y^2=4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point $(4,0)$. Then the equation of the ellipse is:

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is:

If two tangents drawn from a point $P$ to the parabola $y^2=4x$ are at right angles, then the locus of $P$ is:

If the line $2x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3:2$, then $k$ equals:

The length of the diameter of the circle which touches the $x$ -axis at the point $(1,0)$ and passes through the point $(2,3)$ is:

A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ$, where $O$ is the origin. If the area of the triangle $OPQ$ is least, then the slope of the line $PQ$ is:

Locus of centroid of the triangle whose vertices are $(a\cos t,a\sin t)$, $(b\sin t,-b\cos t)$ and $(1,0)$, where $t$ is a parameter, is:

Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is:

The normal to the curve $x=a(\cos \theta +\theta \sin \theta )$, $y=a(\sin \theta -\theta \cos \theta )$ at any point $\theta$ is such that:

The line parallel to the x-axis and passing through the intersection of the lines $ax+2by+3b=0$ and $bx-2ay-3a=0$, where $(a,b)\ne (0,0)$ is:

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

If non-zero numbers $a,b,c$ are in H.P., then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point. That point is:

If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle is:

If the circles $x^2+y^2+2ax+cy+a=0$ and $x^2+y^2-3ax+dy-1=0$ intersect in two distinct points $P$ and $Q$, then the line $5x+by-a=0$ passes through $P$ and $Q$ for:

A circle touches the x-axis and also touches the circle with centre at $(0,3)$ and radius $2$. The locus of the centre of the circle is:

If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=p^2$ orthogonally, then the equation of the locus of its centre is:

An ellipse has $OB$ as semi minor axis, $F$ and $F^{\prime }$ its foci and the angle $\angle FB{F}^{\prime }$ is a right angle. Then the eccentricity of the ellipse is:

If the pair of lines $a{x}^2+2(a+b)xy+b{y}^2=0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then:

If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are both in G.P. with the same common ratio, then the points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$:

If the equation of the locus of a point equidistant from the points $(a_1,b_1)$ and $(a_2,b_2)$ is $(a_1-a_2)x+(b_1-b_2)y+c=0$, then the value of '$c$ ' is:

If the pair of straight lines $x^2-2pxy-y^2=0$ and $x^2-2qxy-y^2=0$ be such that each pair bisects the angle between the other pair, then:

A square of side $a$ lies above the $x$ -axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha$ ($0<\alpha <\pi /4$) with the positive direction of $x$ -axis. The equation of its diagonal not passing through the origin is:

If the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct points, then:

The lines $2x-3y=5$ and $3x-4y=7$ are diameters of a circle having area as 154 sq units. Then the equation of the circle is:

The normal at the point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$, then:

If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is

A point on the parabola $y^2=18x$ at which the ordinate increases at twice the rate of the abscissa is

Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $ABC.$ If the centroid of this triangle moves on the line $2x+3y=1,$ then the locus of the vertex $C$ is the line

The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

The normal to the curve $x=a(1+\cos \theta ),y=a\sin \theta$ at '$\theta$ ' always passes through the fixed point

If one of the lines given by $6x^2-xy+4c{y}^2=0$ is $3x+4y=0,$ then $c$ equals

A variable circle passes through the fixed point $A(p,q)$ and touches $x$ -axis. The locus of the other end of the diameter through $A$ is

If the sum of the slopes of the lines given by $x^2-2cxy-7y^2=0$ is four times their product, then $c$ has the value