Coordinate Geometry PYQ — Page 18

If a $\Delta ABC$ has vertices $A(–1,7),B(–7,1)$ and $C(5,–5)$, then its orthocentre has coordinates:

For some $\theta \in (0,\frac{\pi }{2}),$ if the eccentricity of the hyperbola, ${x}^{2}-{y}^{2}{\mathrm{sec}}^{2}\theta =10$ is $\sqrt{5}$ times the eccentricity of the ellipse, ${x}^{2}{\mathrm{sec}}^{2}\theta +{y}^{2}=5,$ then the length of the latus rectum of the ellipse, is

For $a>0,$ let the curves ${C}_{1}:{y}^{2}=ax$ and ${C}_{2}:{x}^{2}=ay$ intersect at origin $O$ and a point $P.$ Let the line $x=b(0<b<a)$ intersect the chord $OP$ and the $x$ -axis at points $Q$ and $R,$ respectively. If the line $x=b$ bisects the area bounded by the curves, ${C}_{1}$ and ${C}_{2},$ and the area of $\Delta OQR=\frac{1}{2},$ then ‘ $a$ ’ satisfies the equation:

A triangle $ABC$ lying in the first quadrant has two vertices as $A(1,2)$ and $B(3,1)$. If$\angle BAC={90}^{o},$and $ar(\Delta \mathrm{ABC})=5\sqrt{5}$ sq. units, then the abscissa of the vertex $C$ is :

A ray of light coming from the point $(2,2\sqrt{3})$ is incident at an angle $30^{\circ}$ on the line $x=1$ at the point $A$. The ray gets reflected on the line $x=1$ and meets $x$ -axis at the point $B$. Then, the line $AB$ passes through the point

A hyperbola having the transverse axis of length, $\sqrt{2}$ has the same foci as that of the ellipse, $3{x}^{2}+4{y}^{2}=12$ then this hyperbola does not pass through which of the following points?

Two vertices of a triangle are $(0,2)$ and $(4,3).$ If its orthocenter is at the origin, then its third vertex lies in which quadrant?

Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0.$ If its diagonals intersect at $(2,4),$ then one of its vertex is:

Two circles with equal radii are intersecting at the points (0,1) and (0,-1) . The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles 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:

The tangent and the normal lines at the point $(\sqrt{3}, 1)$ to the circle ${x}^{2}+{y}^{2}=4$ and the $x$ -axis form a triangle. The area of this triangle (in square units) is:

The sum of the squares of the lengths of the chords intercepted on the circle, ${x}^{2}+{y}^{2}=16,$ by the lines, $x+y=n, n\in N,$ where $N$ is the set of all natural numbers is:

The locus of the centres of the circles, which touch the circle, ${x}^{2}+{y}^{2}=1$ externally, also touch the $y$-axis and lie in the first quadrant, is:

The length of the chord of the parabola ${x}^{2}=4y$ having equation $x-\sqrt{2}y+4\sqrt{2}=0$ is

Suppose that the points $(h, k), (1, 2)$ and $(-3, 4)$ lie on the line ${L}_{1}.$ If a line ${L}_{2}$ passing through the points $(h, k)$ and $(4, 3)$ is perpendicular to ${L}_{1},$ then $\frac{k}{h}$ equals:

Slope of a line passing through $P(2, 3)$ and intersecting the line $x+y=7$ at a distance of $4$ units from $P,$ is

Lines are drawn parallel to the line $4x-3y+2=0,$ at a distance $\frac{3}{5}$ units from the origin. Then which one of the following points lies on any of these lines?

Let $S={(x,y)\in {R}^{2}:\frac{{y}^{2}}{1+r}-\frac{{x}^{2}}{1-r}=1},$ where $r\neq \pm 1.$ Then $S$ represents:

Let the length of the latus rectum of an ellipse with its major axis along $x$ -axis and centre at the origin, be 8 . If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

Let the equations of two sides of a triangle be $3x-2y+6=0$ and $4x+5y-20=0.$ If the orthocenter of this triangle is at $(1, 1)$ then the equation of it's third side is:

Let $0<\theta <\frac{\pi }{2}.$ If the eccentricity of the hyperbola $\frac{{x}^{2}}{{\mathrm{cos}}^{2}⁡\theta }-\frac{{y}^{2}}{{\mathrm{sin}}^{2}⁡\theta }=1$ is greater than $2,$ then the length of its latus rectum lies in the interval:

Let $S$ be the set of all triangles in the $xy$ -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area $50$ sq. units, then the number of elements in the set $S$ is:

Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola, ${y}^{2}=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this parabola, such that the area of $\Delta PXQ$ is maximum. Then this maximum area (in sq. units) is :

Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then, the locus of a point $P$ such that the perimeter of $\Delta AOP$ is $4$ is