# values-find word problem

• Nov 18th 2007, 01:17 PM
lra11
values-find word problem
1. find the value of k for which y= kx+1 is a tangent to the curve y^2 = 8x

2. the hypotenuse of a triangle measures 17cm. The shortest side is 7cm shorter than the middle side. Find the lengths of the sides.

• Nov 19th 2007, 06:36 AM
kalagota
Quote:

Originally Posted by lra11
1. find the value of k for which y= kx+1 is a tangent to the curve y^2 = 8x
.

y=kx+1 is tangent to $\displaystyle y^2=8x$..
therefore, they have a unique common point, say (a,b), that satisfies both equation.. so $\displaystyle b=ka+1 \implies b^2 = k^2a^2 + 2ka + 1$ and $\displaystyle b^2=8a$
if we equate the $\displaystyle b^2$, we have $\displaystyle k^2a^2 + 2ka + 1 = 8a \implies k^2a^2 + 2ka - 8a + 1 = k^2a^2 + a(2k - 8) + 1 = 0$

this is a quadratic equation hence, we can solve for the value of a:
$\displaystyle a = \frac{8-2k \pm \sqrt{(2k-8)^2 - 4k^2}}{2k^2}$

we want a unique solution for a, so we set the discriminant be equal to 0..

$\displaystyle (2k-8)^2 - 4k^2 = 0$.. solve for k.. Ü

Quote:

Originally Posted by lra11
2. the hypotenuse of a triangle measures 17cm. The shortest side is 7cm shorter than the middle side. Find the lengths of the sides.

since "hypotenuse" came, it can be assumed that this is a right triangle, hence $\displaystyle 17^2 = a^2 + (a-7)^2$.. do the algebra..