# Math Help - simultaneous equations

1. ## simultaneous equations

completely stuck here:

1. find the value of k for which y = 3x+1 is a tangent to the curve
x^2 + y^2 = k

2. find the range of values of k for which y=x-3 meets x^2-3y^2=k in two distinct points.

and i went wrong in this question, no idea where though

find the number of points of intersection
y= x/4 + 1
y^2 = x

i multiplied by 4 to get rid of the fraction. do i square root x ??

2. Hello, lra11

2. Find the range of values of $k$ for which $y\:=\:x-3$
meets $x^2-3y^2\:=\:k$ in two distinct points.
Find their intersections; substitute the first equation into the second.

. . $x^2 - 3(x - 3)^2 \:=\:k\quad\Rightarrow\quad 2x^2 - 8x + (k+27) \:=\:0$

Quadratic Formula: . $x \;=\;\frac{-(-8)\pm\sqrt{(-8)^2 - (4)(2)(k+27)}}{2(2)} \;=\;\frac{8 \pm\sqrt{-8k-152}}{4}$

A quadratic equation has two distinct roots if its Discriminant is positive.

So we have: . $-8k - 152 \:> \:0\quad\Rightarrow\quad -8k \:> \:152\quad\Rightarrow\quad k \:< \:-19$

Therefore, for two distinct intersections: . $k \in (-\infty,\;-19)$