Summary:: The set of values of ##b## for which the origin and the point ##(1, 1)## lie on the same side of the straight line ##a^2x+aby+1=0## ##\forall~a\in\mathbb{R},~b>0##.
(a) ##a\geq1## or ##a\leq-3##
(b) ##a\in~(-3,~0)\cup(\frac13,~1)##
(c) ##a\in~(0,~1)##
(d) ##a\in~(-\infty,~0)##
I...
1.To shift the graph of a function :
Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative.
Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
The problem :
A variable line, drawn through the point of intersection of the straight lines ##(x/a)+(y/b) = 1## and ##(x/b)+(y/a)=1##, meets the coordinate axes in A & B . We have to Show that the locus of the mid point of AB is the curve ##2xy(a + b) = ab(x + y)##.
The solution :
Let ##(h...
Given the coordinates ##P = (3,4)## , find the coordinates of ##P"(x',y')## when the origin is shifted to (1, –2), and the axes are rotated by 90° in the clockwise direction.
I attempted to solve this problem using the following formulas :
##x = X + h## and ##y = Y + k## for translation of the...
Hi, I am really stuck!! I need to find the equation of the plane through the line x=2y=3z perpendicular to the plan 5x+4y-3z=8. Can anyone give me any pointers of where to start with this? Not expecting a full solution, just an idea of where to start.
THanks!
Homework Statement
The line "4x + 3y -4=0 " divides the circumference of circle centred at (5,3) in ratio 1:2.whats the equation of the circle
Homework Equations
2πr
Distance of a line from point
Radius of general circle
The Attempt at a Solution
I tried finding the distance of the centre...
Hello
In India, SL Loney's Elements of Coordinate Geometry is very popular for entrance examinations. I wanted to refresh my coordinate geometry, so tried reading through the book. But I found that the language used is old. I found myself referring to current material on the topic to properly...
Homework Statement
If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0
Homework Equations
Point-slope form
The Attempt at a Solution
I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than...
.The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is
(1) 901 (2) 861 (3) 820 (4) 780
my attempt:
for this to be true i know that sum of x and y coordinate should be 41 but i don't know how to proceed.
Homework Statement
If two vertex of a square of the side AB is A(1,2) and B(2,4) find other two vertex C and D?
Homework Equations
1. y-y1 = m ( x-x1)
2. m=y1-y2 / x1-x2
3. m1*m2= -1
4. (x-x1) / (x1-x2) =(y-y1)/(y1-y2)
5. m=tanA
6. If ax+by+c=0 then its parallel line is ax+by+k=0
The Attempt...
Find the equation of the ellipse whose eccentricity is 2/3 and which has (2,0) and x+y=0 for focus and corresponding directrix .
given answer: (x-2)^2 + y^2 = 4/9 ( ( (x+y)/√2)^2)
what i tried doing:-
ae=2
⇒a*2/3=2
∴a=3
found b=√5.
what to do next? please help.
Homework Statement
if the tangent at a point P("theta") on the ellipse
16 (x^2) + 11 (y^2) = 256
is also tangent to the circle
(x^2) + (y^2) + 2(x) = 15
then ("theta") = ??
2. The attempt at a solution
{{{{ i have taken "theta" as "d" }}}}
P [4 cos d , (16/(sqrt11)) sin d]
equation of...