need help in lines...!!
hi guys!! i really need help in my assignment...can you help me solve this?:
a.) What are the equations of the lines through (-5,-3) and passing at distance two square root of five from (5,7)?
b.)A point moves so that the ratio of its distances from 3x+4y+8=0 and 4x+3y-6=0 is 2. find the equation of its locus.
I really need to answer this today.. please help me...
I assume you mean the perpendicular distance?Originally Posted by april29
A rough diagram will help to visualise the following situation.
Let P(a, b) be the point on the line such that the distance between P and (5, 7) is 2sqrt{5}. Then:
(a - 5)^2 + (b - 7)^2 = 20 .... (1)
The lines obviously have the form y + 3 = m(x + 5).
Also (a, b) is on the line and so:
b + 3 = m(a + 5) .... (2)
The line segment joining P and (5, 7) is perpendicular to the line and so:
(b - 7)/(a - 5) = -1/m .... (3)
Solve simultaneously to get the value of m.
I get m = 2 (for P(1,9)) or m = 1/2 (for P(7,3)) but I sometimes slip up .......
Off-hand, I don't see a more elegant way.
I assume you mean distance from 3x+4y+8=0 is twice distance from 4x+3y-6=0.
Here are two hints to get you started:
1. The locus passess through the point of intersection of 3x+4y+8=0 and 4x+3y-6=0 (do you see why?)
2. The locus consists of TWO lines (do you see why?) and they both pass through the aforementioned intersection point.
But now I've just thought of one ......
Find the values of m such that the line y + 3 = m(x + 5) <=> y = mx + 5m - 3 is tangent to the circle with centre (5, 7) and radius 2 sqrt{5}, that is, the circle. (What circle geometry theorem does this approach exploit?)
To do this, set up the equation that gives the x-coordinates of the intersection points of the line with the circle by substituting y = mx + 5m - 3 into
You get a quadratic in x.
Set the discriminant (which will be a function of m) of this quadratic equal to zero and solve for m.
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