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Topic
First Poster
Last Poster
slope product and area of triangle
smartvong 1
N
by Lankou
Let 
be the origin, 
, and 
. The product of the slopes of the lines 
and 
is 4. The trajectory of the moving point 
is denoted by 
.
(1) Find the equation of.
(2) A line
passes through the point 
and intersects at points 
and 
. The midpoint 
of segment 
lies in the first quadrant with a vertical coordinate of 
. Find the area of 
.
be the origin, 
, and 
. The product of the slopes of the lines 
and 
is 4. The trajectory of the moving point 
is denoted by 
.(1) Find the equation of
.(2) A line
passes through the point 
and intersects at points 
and 
. The midpoint 
of segment 
lies in the first quadrant with a vertical coordinate of 
. Find the area of 
.
1 reply
Integer points on the line
Silverfalcon 8
N
by SomeonecoolLovesMaths
A line passes through 
and 
. How many other points with integer coordinates are on the line and strictly between 
and 
?
and 
. How many other points with integer coordinates are on the line and strictly between 
and 
?
8 replies
[PMO26 Qualifying II.13] Three Lines (That) Form A Triangle
kae_3 0
Three lines with slope 
, 
, and 
intersect to form a nondegenerate isosceles triangle. If 
, what is ?
, 
, and 
intersect to form a nondegenerate isosceles triangle. If 
, what is ?
0 replies
Line of slope i
eduD_looC 1
N
by MineCuber
What happens when you rotate a line of slope 
by 
degrees? All I've got is that if 
, then the new line should still have slope since 
. :/
On a second thought, is it even possible to rotate such a line?
I remember reading about this because "five points determine a conic" and "three points are sufficient to determine a circle". So there are two special points on all circles (smth like that).
by 
degrees? All I've got is that if 
, then the new line should still have slope since 
. :/On a second thought, is it even possible to rotate such a line?
I remember reading about this because "five points determine a conic" and "three points are sufficient to determine a circle". So there are two special points on all circles (smth like that).
1 reply
A Certain Function
4everwise 4
N
by RedFireTruck
A certain function 
has the properties that 
for all positive real values of 
, and that 
for 
. Find the smallest for which 
.
has the properties that 
for all positive real values of 
, and that 
for 
. Find the smallest for which 
.
4 replies
conics time :(
am07 2
N
by AndrewTom
A normal with slope 
is drawn from the point 
to the parabola 
where 
Let 
be the line passing through and parallel to the directrix of the parabola. Suppose that intersects the parabola at two points 
and 
Let 
denote the length of the latus rectum and 
denote the square of the length of the line segment 
If 
then the value of 
is ______.
is drawn from the point 
to the parabola 
where 
Let 
be the line passing through and parallel to the directrix of the parabola. Suppose that intersects the parabola at two points 
and 
Let 
denote the length of the latus rectum and 
denote the square of the length of the line segment 
If 
then the value of 
is ______.
2 replies
Convex function trouble
Sedro 2
N
by Sedro
Suppose a function satisfies 
for all real 
. Then, we have 
We also have ![\[f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2}) \ge 2f(x).\]](//latex.artofproblemsolving.com/5/3/d/53d5e6a96b03a5c4a05a9da1945cbaa81829271e.png)
Adding the first two inequalities and using the third, we have ![\[2f(x) + f(y)+f(-y) \ge 2(f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2})) \ge 4f(x),\]](//latex.artofproblemsolving.com/a/9/c/a9c9255deeb88e31603a55d6cf32fbf864ca0758.png)
which implies 
. But this means is bounded above which is absurd -- take the identity function. I feel I am making some blithely stupid mistake, but I can't see what. Any help is appreciated :)
satisfies 
for all real 
. Then, we have 
We also have ![\[f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2}) \ge 2f(x).\]](http://latex.artofproblemsolving.com/5/3/d/53d5e6a96b03a5c4a05a9da1945cbaa81829271e.png)
Adding the first two inequalities and using the third, we have ![\[2f(x) + f(y)+f(-y) \ge 2(f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2})) \ge 4f(x),\]](http://latex.artofproblemsolving.com/a/9/c/a9c9255deeb88e31603a55d6cf32fbf864ca0758.png)
which implies 
. But this means is bounded above which is absurd -- take the identity function. I feel I am making some blithely stupid mistake, but I can't see what. Any help is appreciated :)
2 replies
Could you find the reference of this problem? pic? GEOMETRY
KR_Geometry 1
N
by vanstraelen
Incircle is attached on ABCD
And the slope of AB is root3
And the slope of AB is root3
1 reply
Student F back at it again
fruitmonster97 1
N
by AndrewTom
Student F is in a trig class, but they missed the first three classes, so when asked for the value of the tangent of 
in homework, they assume the problem is asking for the y-intercept of the tangent line of a circle centered at the origin with area 
with slope 
Assuming they got their intepretation of the problem correct, what is the difference between their answer and the correct answer?
in homework, they assume the problem is asking for the y-intercept of the tangent line of a circle centered at the origin with area 
with slope 
Assuming they got their intepretation of the problem correct, what is the difference between their answer and the correct answer?
1 reply
Geometry problem
Rice_Farmer 3
N
by Rice_Farmer
Let and 
be real numbers with 
such that the value of 
is maximized. Find 
and 
be real numbers with 
such that the value of 
is maximized. Find 
3 replies
