Coordinate Geometry · NDA previous year questions

If p and q are real numbers between 0 and 1 such that the points (p,1), (1,q) and (0,0) form an equilateral triangle, then what is (p+q) equal to?

1. A.$\sqrt{2}$
2. B.$\sqrt{2}-1$
3. C.$2-\sqrt{3}$
4. D.$4-2\sqrt{3}$

The vertices of a triangle are A(1, 1),B(0, 0) and C(2, 0). The angular bisectors of the triangle meet at P. What are the coordinates of P?

1. A.$(1,\sqrt{2}-1)$
2. B.$(1,\sqrt{3}-1)$
3. C.(1, 1/2)
4. D.$(1/2,\sqrt{2}-1)$

Let A(3, -1) and B(1, 1) be the end points of line segment AB. Let P be the middle point of the line segment AB. Let Q be the point situated at a distance $\sqrt{2}$ units from P on the perpendicular bisector line of AB. What are the possible coordinates of Q?

1. A.(2,1)
2. B.(3,1)
3. C.(2,2)
4. D.(1,3)

ABC is an equilateral triangle and AD is the altitude on BC. If the coordinates of A are (1,2) and that of D are (−2,6), then what is the equation of BC?

1. A.3x+4y−18=0
2. B.4x+3y−1=0
3. C.4x−3y+26=0
4. D.3x−4y+30=0

What is the equation of the circle whose diameter is 10 cm and the equations of two of its diameters are x+y=0 and x−y=0?

1. A.$x^2+y^2=1$
2. B.$x^2+y^2=25$
3. C.$x^2+y^2=100$
4. D.$x^2+y^2-2x-2y-23=0$

A square is inscribed in a circle $x^2+y^2+2x+2y+1=0$ and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?

1. A.(-2,2)
2. B.(-2,-2)
3. C.$(-1+\frac{1}{\sqrt{2}},-1-\frac{1}{\sqrt{2}})$
4. D.None of the above

A tangent to the parabola $y^2=4x$ is inclined at an angle $45^{\circ }$ deg with the positive direction of x-axis. What is the point of contact of the tangent and the parabola?

1. A.(1, 1)
2. B.$(2,2\sqrt{2})$
3. C.$(\frac{1}{2},\frac{1}{\sqrt{2}})$
4. D.(1,2)

. What is the distance between the two foci of the hyperbola $25x^2-75y^2=225$ ?

1. A.$2\sqrt{3}\mathrm{units}$
2. B.$4\sqrt{3}\mathrm{units}$
3. C.$\sqrt{6}\mathrm{units}$
4. D.$2\sqrt{6}\mathrm{units}$

If any point on an ellipse is $(3\sin \alpha ,5\cos \alpha )$, then what is the eccentricity of the ellipse?

1. A.4/3
2. B.4/5
3. C.3/4
4. D.1/2

A(1,2,−1), B(2,5,−2) and C(4,4,−3) are three vertices of a rectangle. What is the area of the rectangle?

1. A.8 square units
2. B.9 square units
3. C.$\sqrt{66}$ square units
4. D.$\sqrt{68}$ square units

ABC is a triangle right-angled at B. If A(k,1,−1), B(2k,0,2) and C(2+2k,k,1) are the vertices of the triangle, then what is the value of k?

1. A.-3
2. B.-1
3. C.1
4. D.3

If a line $\frac{x+1}{p}=\frac{y-1}{q}=\frac{z-2}{r}$ where p=2q=3r, makes an angle $\theta$ with the positive direction of y-axis, then what is $\cos 2\theta$ equal to?

1. A.-31/49
2. B.-37/49
3. C.31/49
4. D.37/49

The foci of the ellipse $4x^2+9y^2=1$ are at Q and R. If P(x, y) is any point on the ellipse, then what is PQ + PR equal to?

1. A.2
2. B.1
3. C.2/3
4. D.1/3

If P(2, 4), Q(8, 12), R(10, 14) and S(x, y) are vertices of a parallelogram, then what is (x + y) equal to?

1. A.8
2. B.10
3. C.12
4. D.14

The equation of a circle is $(x^2-4x+3)+(y^2-6y+8)=0$ Which of the following statements are correct? I. The end points of a diameter of the circle are at (1, 2) and (3, 4). II. The end points of a diameter of the circle are at (1, 4) and (3, 2). III. The end points of a diameter of the circle are at (2, 4) and (4, 2). Select the answer using the code given below.

1. A.I and II only
2. B.II and III only
3. C.I and III only
4. D.I, II and III

Consider the points P(4k, 4k) and Q(4k, -4k) lying on the parabola $y^2=4kx$. If the vertex is A, then what is $\angle \mathrm{PAQ}$ equal to?

1. A.$60^{\circ }$
2. B.$90^{\circ }$
3. C.$120^{\circ }$
4. D.$135^{\circ }$

Direction: Consider the following for the items that follow: A triangle ABC is inscribed in the circle $x^2+y^2=100$. B and C have coordinates (6, 8) and (-8, 6) respectively. What is BAC equal to?

1. A.$\pi /2$
2. B.$\pi /3$ or $2\pi /3$
3. C.$\pi /4$ or $3\pi /4$
4. D.$\pi /6$ or $5\pi /6$

Direction: Consider the following for the items that follow: ABCD is an isosceles trapezium and AB is parallel to DC. Let A(2, 3), B(4, 3), C(5, 1) be the vertices. What are the coordinates of vertex D?

1. A.(2, 1)
2. B.(1, 2)
3. C.(1, 1)
4. D.(3, 1)

What is the area of the parabola bounded by the latus rectum?

1. A.1/6 square unit
2. B.2/3 square unit
3. C.1 square unit
4. D.4/3 square units

A triangle $PQR$ is such that 3 points lie on the side $PQ$, 4 points on $QR$ and 5 points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

1. A.205
2. B.206
3. C.215
4. D.220

If ${\log }_{b}a=p$, ${\log }_{d}c=2p$ and ${\log }_{f}e=3p$, then what is $(ace{)}^{\frac{1}{p}}$ equal to ?

1. A.$b{d}^2{f}^3$
2. B.$bdf$
3. C.$b^3{d}^{2}f$
4. D.$b^2{d}^2{f}^2$

If (a, b) is the centre and c is the radius of the circle $x^2+y^2+2x+6y+1=0$, then what is the value of $a^2+b^2+c^2$?

1. A.19
2. B.18
3. C.17
4. D.11

If a variable line passes through the point of intersection of the lines $x+2y-1=0$ and $2x-y-1=0$ and meets the coordinate axes in $A$ and $B$, then what is the locus of the mid-point of $AB$?

1. A.$3x+y=10xy$
2. B.$x+3y=10xy$
3. C.$3x+y=10$
4. D.$x+3y=10$

What is the equation to the straight line passing through the point $(-\sin \theta ,\cos \theta )$ and perpendicular to the line $x\cos \theta +y\sin \theta =9$?

1. A.$x\sin \theta -y\cos \theta -1=0$
2. B.$x\sin \theta -y\cos \theta +1=0$
3. C.$x\sin \theta -y\cos \theta =0$
4. D.$x\cos \theta -y\sin \theta +1=0$

Two points $P$ and $Q$ lie on line $y=2x+3$. These two points $P$ and $Q$ are at a distance 2 units from another point $R(1,5)$. What are the coordinates of the points $P$ and $Q$?

1. A.$(1+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})$
2. B.$(3+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(-1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})$
3. C.$(1-\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(1+\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})$
4. D.$(3-\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(-1+\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})$