Coordinate Geometry · NDA previous year questions

For the equation $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$ where $a\ne 0$, to represent a circle, the condition will be

1. A.$a=b$ and $c=0$
2. B.$f=g$ and $h=0$
3. C.$a=b$ and $h=0$
4. D.$f=g$ and $c=0$

What is the locus of a point which moves equidistant from the coordinate axes?

1. A.$x\pm y=0$
2. B.$x+2y=0$
3. C.$2x+y=0$
4. D.None of the above

What is the equation to the straight line joining the origin to the point of intersection of the lines $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ ?

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

What is the equation to circle which touches both the axes and has centre on the line $x+y=4$?

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

If the straight lines $x-2y=0$ and $kx+y=1$ intersect at the point $(1,\frac{1}{2})$, then what is the value of $k$?

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

A point $P$ moves such that the difference of its distances from two given points $(c,0)$ and $(-c,0)$ is constant. What is the locus of the point $P$?

1. A.Circle
2. B.Ellipse
3. C.Hyperbola
4. D.Parabola

Under which of the following conditions does a general second-degree equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ ($a\ne 0$) represent a circle?

1. A.$h=g,a=b$
2. B.$h=g=f,a=b$
3. C.$h=0,a=b$
4. D.$h=0,g^2+f^2-c=a+b$

What is the area of a triangle whose vertices are at $(3,-1,2)$, $(1,-1,-3)$ and $(4,-3,1)$?

1. A.$\frac{\sqrt{165}}{2}$
2. B.$\frac{\sqrt{135}}{2}$
3. C.4
4. D.2

If the latus rectum of an ellipse is equal to half its minor axis, then what is its eccentricity?

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

If $A=\{2,3\},B=\{4,5\},C=\{5,6\}$, then what is the number of elements in $A\times (B\cap C)$?

1. A.2
2. B.4
3. C.6
4. D.8

What is the slope of the line perpendicular to the line $\frac{x}{4}+\frac{y}{3}=1$ ?

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

$ABC$ is a triangle in which $BC=10\mathrm{cm}$, $CA=6\mathrm{cm}$ and $AB=8\mathrm{cm}$. Which one of the following is correct?

1. A.$ABC$ is an acute-angled triangle
2. B.$ABC$ is an obtuse-angled triangle
3. C.$ABC$ is a right-angled triangle
4. D.None of the above

If the AM and GM between two numbers are in the ratio $m:n$, then what is the ratio between the two numbers?

1. A.$\frac{m+\sqrt{m^2-n^2}}{m-\sqrt{m^2-n^2}}$
2. B.$\frac{m+n}{m-n}$
3. C.$\frac{m^2-n^2}{m^2+n^2}$
4. D.$\frac{m^2+n^2-mn}{m^2+n^2+mn}$

If the area of a triangle with vertices $(-3,0)$, $(3,0)$ and $(0,k)$ is $9$ square units, then what is the value of $k$?

1. A.$3$
2. B.$6$
3. C.$9$
4. D.$12$

A circle is drawn with the two foci of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the end of the diameter. What is the equation to the circle?

1. A.$x^2+y^2=a^2+b^2$
2. B.$x^2+y^2=a^2-b^2$
3. C.$x^2+y^2=2(a^2+b^2)$
4. D.$x^2+y^2=2(a^2-b^2)$

What is the image of the point $(1,2)$ on the line $3x+4y-1=0$?

1. A.$(-\frac{7}{5},-\frac{6}{5})$
2. B.$(\frac{7}{8},\frac{1}{2})$
3. C.$(\frac{7}{8},-\frac{1}{2})$
4. D.$(-\frac{7}{5},\frac{1}{2})$

If $(4,0)$ and $(-4,0)$ are the foci of an ellipse and the semi-minor axis is 3, then the ellipse passes through which one of the following points?

1. A.$(2,0)$
2. B.$(0,5)$
3. C.$(0,0)$
4. D.$(5,0)$

What are the equations of the directrices of the ellipse $25x^2+16y^2=400$?

1. A.$3x\pm 25=0$
2. B.$3y\pm 25=0$
3. C.$x\pm 15=0$
4. D.$y\pm 25=0$

Let $E$ be the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and $C$ be the circle $x^2+y^2=9$. Let $P=(1,2)$ and $Q=(2,1)$. Which one of the following is correct?

1. A.$Q$ lies inside $C$ but outside $E$
2. B.$Q$ lies outside both $C$ and $E$
3. C.$P$ lies inside both $C$ and $E$
4. D.$P$ lies inside $C$ but outside $E$

What is the geometric interpretation of the identity $(\vec{a}-\vec{b})\times (\vec{a}+\vec{b})=2(\vec{a}\times \vec{b})$? 1. If the diagonals of a given parallelogram are used as sides of a second parallelogram, then the area of the second parallelogram is twice that of the given parallelogram. 2. If the semi-diagonals of a given parallelogram are used as sides of a second parallelogram, then the area of the second parallelogram is half that of the given parallelogram. Select the correct answer using the code given below:

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is ${\int }_0^{\pi /2}\frac{{\sin }^{3}x}{{\sin }^{3}x+{\cos }^{3}x}dx$?

1. A.$\pi$
2. B.$\frac{\pi }{2}$
3. C.$\frac{\pi }{4}$
4. D.0

What is the area of the triangle with vertices (0, 2, 2), (2, 0, -1) and (3, 4, 0)?

1. A.15/2 square units
2. B.15 square units
3. C.7/2 square units
4. D.7 square units

If $\frac{1}{2-\sqrt{-2}}$ is one of the roots of $a{x}^2+bx+c=0$ where a, b, c are real, then what are the values of a, b, c respectively?

1. A.6, -4, 1
2. B.4, 6, -1
3. C.3, -2, 1
4. D.6, 4, 1

Let O (0, 0, 0), P (3, 4, 5), Q (m, n, r) and R (1, 1, 1) be the vertices of a parallelogram taken in order. What is the value of $m+n+r$?

1. A.6
2. B.12
3. C.15
4. D.More than 15

What is the locus of points, the difference of whose distances from two points being constant?

1. A.Pair of straight lines
2. B.An ellipse
3. C.A hyperbola
4. D.A parabola