Coordinate Geometry · NDA previous year questions

For the next two (2) items that follow : Consider the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ If the circles intersect at two distinct points, then which one of the following is correct?

1. A.$r=1$
2. B.$1<r<2$
3. C.$r=2$
4. D.$2<r<8$

For the next two (2) items that follow: Consider a circle passing through the origin and the points $(a,b)$ and $(-b,-a)$. On which line does the centre of the circle lie?

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

For the next two (2) items that follow: Consider a circle passing through the origin and the points $(a,b)$ and $(-b,-a)$. What is the sum of the squares of the intercepts cut off by the circle on the axes?

1. A.${\left(\frac{a^2+b^2}{a^2-b^2}\right)}^2$
2. B.$2{\left(\frac{a^2+b^2}{a-b}\right)}^2$
3. C.$4{\left(\frac{a^2+b^2}{a-b}\right)}^2$
4. D.None of the above

If a circle of radius $b$ units with centre at $(0,b)$ touches the line $y=x-\sqrt{2}$, then what is the value of $b$?

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

For the next two (2) items that follow : Consider the curves $f(x)=x|x|-1$ and $g(x)=\{\begin{array}{ll}\frac{3x}{2},& x>0\\ 2x,& x\le 0\end{array}$ Where do the curves intersect?

1. A.At $(2,3)$ only
2. B.At $(-1,-2)$ only
3. C.At $(2,3)$ and $(-1,-2)$
4. D.Neither at $(2,3)$ nor at $(-1,-2)$

$z\bar{z}+(3-i)z+(3+i)\bar{z}+1=0$ represents a circle with

1. A.centre $(-3,-1)$ and radius 3
2. B.centre $(-3,1)$ and radius 3
3. C.centre $(-3,-1)$ and radius 4
4. D.centre $(-3,1)$ and radius 4

The value of the infinite product $6^{\frac{1}{2}}\times 6^{\frac{1}{2}}\times 6^{\frac{3}{8}}\times 6^{\frac{1}{4}}\times \dots$ is

1. A.6
2. B.36
3. C.216
4. D.$\infty$

The $n$th term of an A.P. is $\frac{3+n}{4}$, then the sum of first 105 terms is

1. A.270
2. B.735
3. C.1409
4. D.1470

The line $3x+4y-24=0$ intersects the $x$-axis at $A$ and $y$-axis at $B$. Then the circumcentre of the triangle $OAB$ where $O$ is the origin is

1. A.$(2,3)$
2. B.$(3,3)$
3. C.$(4,3)$
4. D.None of the above

The product of the perpendiculars from the two points $(\pm 4,0)$ to the line $3x\cos \varphi +5y\sin \varphi =15$ is

1. A.25
2. B.16
3. C.9
4. D.8

Two straight lines passing through the point $A(3,2)$ cut the line $2y=x+3$ and $x$-axis perpendicularly at $P$ and $Q$ respectively. The equation of the line $PQ$ is

1. A.$7x+y-21=0$
2. B.$x+7y+21=0$
3. C.$2x+y-8=0$
4. D.$x+2y+8=0$

A straight line $x=y+2$ touches the circle $4(x^2+y^2)=r^2$. The value of $r$ is

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

If the centre of the circle passing through the origin is $(3,4)$, then the intercepts cut off by the circle on $x$-axis and $y$-axis respectively are

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

Consider any point $P$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ in the first quadrant. Let $r$ and $s$ represent its distances from $(4,0)$ and $(-4,0)$ respectively, then $(r+s)$ is equal to

1. A.10 unit
2. B.9 unit
3. C.8 unit
4. D.6 unit

The eccentricity of the hyperbola $16x^2-9y^2=1$ is

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

The lines $2x=3y=-z$ and $6x=-y=-4z$

1. A.are perpendicular
2. B.are parallel
3. C.intersect at an angle $45^{\circ }$
4. D.intersect at an angle $60^{\circ }$

For the next three (3) items that follow: Consider the triangle $ABC$ with vertices $A(-2,3),B(2,1)$ and $C(1,2)$. What is the circumcentre of the triangle $ABC$?

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

For the next three (3) items that follow: Consider the triangle $ABC$ with vertices $A(-2,3),B(2,1)$ and $C(1,2)$. What is the centroid of the triangle $ABC$?

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

For the next three (3) items that follow: Consider the triangle $ABC$ with vertices $A(-2,3),B(2,1)$ and $C(1,2)$. What is the foot of the altitude from the vertex $A$ of the triangle $ABC$?

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

The perpendicular distance between the straight lines $6x+8y+15=0$ and $3x+4y+9=0$ is

1. A.3/2 units
2. B.3/10 unit
3. C.3/4 unit
4. D.2/7 unit

The area of a triangle, whose vertices are $(3,4),(5,2)$ and the point of intersection of the lines $x=a$ and $y=5$, is 3 square units. What is the value of $a$?

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

The length of perpendicular from the origin to a line is 5 units and the line makes an angle $120^{\circ }$ with the positive direction of $x$-axis. The equation of the line is

1. A.$x+\sqrt{3}y=5$
2. B.$\sqrt{3}x+y=10$
3. C.$\sqrt{3}x-y=10$
4. D.None of the above

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

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

A line passes through $(2,2)$ and is perpendicular to the line $3x+y=3$. Its $y$-intercept is

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

The point on the parabola $y^2=4ax$ nearest to the focus has its abscissa

1. A.$x=0$
2. B.$x=a$
3. C.$x=a/2$
4. D.$x=2a$