Coordinate Geometry · NDA previous year questions

What is the equation of the ellipse having foci $(\pm 2,0)$ and the eccentricity $\frac{1}{4}$?

1. A.$\frac{x^2}{64}+\frac{y^2}{60}=1$
2. B.$\frac{x^2}{60}+\frac{y^2}{64}=1$
3. C.$\frac{x^2}{20}+\frac{y^2}{24}=1$
4. D.$\frac{x^2}{24}+\frac{y^2}{20}=1$

$(0,0,0)$, $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$ are four distinct points. What are the coordinates of the point which is equidistant from the four points?

1. A.$\left(\frac{a+b+c}{3},\frac{a+b+c}{3},\frac{a+b+c}{3}\right)$
2. B.$(a,b,c)$
3. C.$\left(\frac{a}{2},\frac{b}{2},\frac{c}{2}\right)$
4. D.$\left(\frac{a}{3},\frac{b}{3},\frac{c}{3}\right)$

Let $ABCD$ be a parallelogram whose diagonals intersect at $P$ and let $O$ be the origin. What is $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$ equal to?

1. A.$2\overrightarrow{OP}$
2. B.$4\overrightarrow{OP}$
3. C.$6\overrightarrow{OP}$
4. D.$8\overrightarrow{OP}$

$ABCD$ is a quadrilateral whose diagonals are $AC$ and $BD$. Which one of the following is correct?

1. A.$\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{AC}+\overrightarrow{DB}$
2. B.$\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{BD}+\overrightarrow{CA}$
3. C.$\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{AC}+\overrightarrow{BD}$
4. D.$\overrightarrow{BA}+\overrightarrow{CD}=\overrightarrow{BC}+\overrightarrow{AD}$

If $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{b}\times \vec{c}=\vec{a}$, then which one of the following is correct?

1. A.$\vec{a},\vec{b},\vec{c}$ are orthogonal in pairs and $|\vec{a}|=|\vec{c}|$ and $|\vec{b}|=1$
2. B.$\vec{a},\vec{b},\vec{c}$ are non-orthogonal to each other
3. C.$\vec{a},\vec{b},\vec{c}$ are orthogonal in pairs but $|\vec{a}|\ne |\vec{c}|$
4. D.$\vec{a},\vec{b},\vec{c}$ are orthogonal in pairs but $|\vec{b}|\ne 1$

If $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$ and $\vec{b}=3\hat{i}+2\hat{j}-\lambda \hat{k}$ are perpendicular, then what is the value of $\lambda$?

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

An equilateral triangle has one vertex at $(0,0)$ and another at $(3,\sqrt{3})$. What are the coordinates of the third vertex?

1. A.$(0,2\sqrt{3})$ only
2. B.$(3,-\sqrt{3})$ only
3. C.$(0,2\sqrt{3})$ or $(3,-\sqrt{3})$
4. D.Neither $(0,2\sqrt{3})$ nor $(3,-\sqrt{3})$

What is the equation of the right bisector of the line segment joining (1, 1) and (2, 3)?

1. A.$2x+4y-11=0$
2. B.$2x-4y-5=0$
3. C.$2x-4y-11=0$
4. D.$x-y+1=0$

If the point (a, a) lies between the lines $|x+y|=2$, then which one of the following is correct?

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

What is the equation of the straight line which passes through the point of intersection of the straight lines $x+2y=5$ and $3x+7y=17$ and is perpendicular to the straight line $3x+4y=10$?

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

If (a, b) is at unit distance from the line $8x+6y+1=0$, then which of the following conditions are correct? 1. $3a-4b-4=0$ 2. $8a+6b+11=0$ 3. $8a+6b-9=0$ Select the correct answer using the code given below :

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

A straight line cuts off an intercept of 2 units on the positive direction of x-axis and passes through the point (-3, 5). What is the foot of the perpendicular drawn from the point (3, 3) on this line?

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

What is the radius of the circle passing through the point $(2,4)$ and having centre at the intersection of the lines $x-y=4$ and $2x+3y+7=0$?

1. A.3 units
2. B.5 units
3. C.$3\sqrt{3}$ units
4. D.$5\sqrt{2}$ units

What is the equation of the hyperbola having latus rectum and eccentricity 8 and $\frac{3}{\sqrt{5}}$ respectively?

1. A.$\frac{x^2}{25}-\frac{y^2}{20}=1$
2. B.$\frac{x^2}{40}-\frac{y^2}{20}=1$
3. C.$\frac{x^2}{40}-\frac{y^2}{30}=1$
4. D.$\frac{x^2}{30}-\frac{y^2}{25}=1$

If the ellipse $9x^2+16y^2=144$ intercepts the line $3x+4y=12$, then what is the length of the chord so formed?

1. A.5 units
2. B.6 units
3. C.8 units
4. D.10 units

What is the eccentricity of rectangular hyperbola?

1. A.$\sqrt{2}$
2. B.$\sqrt{3}$
3. C.$\sqrt{5}$
4. D.$\sqrt{6}$

Consider the following for the next two (02) items that follow: Consider the parabola $y=x^2+7x+2$ and the straight line $y=3x-3$. What are the coordinates of the point on the parabola which is closest to the straight line?

1. A.$(0,2)$
2. B.$(-2,-8)$
3. C.$(-7,2)$
4. D.$(1,10)$

Consider the following for the next two (02) items that follow: Consider the parabola $y=x^2+7x+2$ and the straight line $y=3x-3$. What is the shortest distance from the above point on the parabola to the line?

1. A.$\frac{\sqrt{10}}{2}$
2. B.$\frac{\sqrt{10}}{5}$
3. C.$\frac{1}{\sqrt{10}}$
4. D.$\frac{\sqrt{5}}{4}$

What is the curve which passes through the point $(1,1)$ and whose slope is $\frac{2y}{x}$?

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

If $xdy=ydx+y^{2}dy$, $y>0$ and $y(1)=1$, then what is $y(-3)$ equal to?

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

For the next three (3) items that follow : Consider a parallelogram whose vertices are $A(1,2),B(4,y),C(x,6)$ and $D(3,5)$ taken in order. What is the value of $A{C}^2-B{D}^2$?

1. A.25
2. B.30
3. C.36
4. D.40

For the next three (3) items that follow : Consider a parallelogram whose vertices are $A(1,2),B(4,y),C(x,6)$ and $D(3,5)$ taken in order. What is the point of intersection of the diagonals?

1. A.$\left(\frac{7}{2},4\right)$
2. B.$(3,4)$
3. C.$\left(\frac{7}{2},5\right)$
4. D.$(3,5)$

For the next three (3) items that follow : Consider a parallelogram whose vertices are $A(1,2),B(4,y),C(x,6)$ and $D(3,5)$ taken in order. What is the area of the parallelogram?

1. A.$\frac{7}{2}$ square units
2. B.$4$ square units
3. C.$\frac{11}{2}$ square units
4. D.$7$ square units

A straight line intersects $x$ and $y$ axes at $P$ and $Q$ respectively. If $(3,5)$ is the middle point of $PQ$, then what is the area of the triangle $OPQ$?

1. A.12 square units
2. B.15 square units
3. C.20 square units
4. D.30 square units

For the next two (2) items that follow : Consider the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ What is the distance between the centres of the two circles?

1. A.5 units
2. B.6 units
3. C.8 units
4. D.10 units