This is a question by declon.
Originally Posted by declon
The line y = 4x + k is a tangent to the curve y = x^2 + 4x + 3 at a point P.
(b) Find the value of k.
(c) Find an equation of the normal to the curve y = x^2 + 4x + 3 at the point P.
Ok, let's think this through. A diagram often helps your thinking.
Draw a quick sketch of the parabola y = x^2 + 4x + 3. That is concave up and cuts y-axis at 3. Don't worry about x-intercepts.
Now draw a line (upward sloping as y = 4x + k has a gradient of +4) which touches the parabola at P.
Now the slope at P on the parabola is the same as the slope at P on the straight line. Yes? And that slope is 4. Right?
Now the slope of the parabola is given by the derivative of x^2 + 4x + 3.
This derivative is ................... and it is equal to ............ So you can work out the x-coordinate of P. Use it to find the y-coordinate of P.
(b) Now find the equation of the tangent at P and hence find k.
(c)You've got the point P, you can easily work out the slope of the normal, and use them to find the equation of the normal.