Find the equation of a circle whose centre is $(3, -1)$ and which cuts off a chord of length 6 units on the line $2x - 5y + 18 = 0$.
Perpendicular distance from centre to line $d = \sqrt{29}$. Radius $r^2 = d^2 + (6/2)^2 = 29 + 9 = 38$. Equation: $(x-3)^2 + (y+1)^2 = 38$.
$x^2 + y^2 - 6x + 2y - 28 = 0$
Q4
Find the equation of a circle of radius 5 which is touching another circle $x^2 + y^2 - 2x - 4y - 20 = 0$ at $(5, 5)$.
Given circle centre $(1, 2)$, radius 5. Required circle has radius 5 and touches at $(5, 5)$. Point $(5, 5)$ is midpoint of centres (external touch). Centre $(9, 8)$. Equation: $(x-9)^2 + (y-8)^2 = 25$.
$x^2 + y^2 - 18x - 16y + 120 = 0$
Q5
Find the equation of a circle passing through the point $(7, 3)$ having radius 3 units and whose centre lies on the line $y = x - 1$.
Find the equation of each of the following parabolas: (a) Directrix $x = 0$, focus at $(6, 0)$ (b) Vertex at $(0, 4)$, focus at $(0, 2)$ (c) Focus at $(-1, -2)$, directrix $x - 2y + 3 = 0$
The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 + 6x + 2y = 0$.
Centre $(-3, -1)$. $-3 + 3(-1) = -6 \neq 0$.
$False$
Q12
The shortest distance from the point $(2, -7)$ to the circle $x^2 + y^2 - 14x - 10y - 151 = 0$ is equal to 5.
Centre $(7, 5)$, radius 15. Distance to point is 13. Shortest distance $15-13=2$.
$False$
Q13
If the line $lx + my = 1$ is a tangent to the circle $x^2 + y^2 = a^2$, then the point $(l, m)$ lies on a circle.
Distance from origin to line is $a$. $\frac{1}{\sqrt{l^2+m^2}} = a \implies l^2+m^2 = 1/a^2$.
$True$
Q14
The point $(1, 2)$ lies inside the circle $x^2 + y^2 - 2x + 6y + 1 = 0$.
$1+4-2+12+1 = 16 > 0$. Point is outside.
$False$
Q15
The line $lx + my + n = 0$ will touch the parabola $y^2 = 4ax$ if $ln = am^2$.
Condition $c = a/m'$ where $y=m'x+c$. Here $m'=-l/m, c=-n/m$. $-n/m = a/(-l/m) \implies nl = am^2$.
$True$
Q16
If P is a point on the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1$ whose foci are S and S', then $PS + PS' = 8$.
$2a = 2(5) = 10$.
$False$
Q17
The line $2x + 3y = 12$ touches the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 2$ at the point $(3, 2)$.
Tangent at $(3, 2)$ to $x^2/18 + y^2/8 = 1$ is $3x/18 + 2y/8 = 1 \implies x/6 + y/4 = 1$.
$True$
Q18
The locus of the point of intersection of lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ for different value of k is a hyperbola whose eccentricity is 2.
The equation of the circle having centre at $(3, -4)$ and touching the line $5x + 12y - 12 = 0$ is ________________ .
Radius $r = 45/13$.
$(x-3)^2 + (y+4)^2 = \frac{2025}{169}$
Q20
The equation of the circle circumscribing the triangle whose sides are the lines $y = x + 2, 3y = 4x, 2y = 3x$ is ________________ .
Vertices $(0,0), (4,6), (6,8)$.
$x^2 + y^2 - 46x + 22y = 0$
Q21
An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the length of the string and distance between the pins are ____________.
$2a=6, 2b=4, c=\sqrt{5}$. String length $2a+2c$.
$6 + 2\sqrt{5}$ cm, $2\sqrt{5}$ cm
Q22
The equation of the ellipse having foci $(0, 1), (0, -1)$ and minor axis of length 1 is ________________ .
$c=1, b=1/2, a^2=5/4$.
$20x^2 + 4y^2 = 5$
Q23
The equation of the parabola having focus at $(-1, -2)$ and the directrix $x - 2y + 3 = 0$ is ________________ .
Using definition.
$4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0$
Q24
The equation of the hyperbola with vertices at $(0, \pm 6)$ and eccentricity $\frac{5}{3}$ is ______________ and its foci are ________________ .