To find the value of* k*, you need to first **find the gradient of the tangent** of the curve. Then you will **differentiate the equation** of the curve and **equate it to the gradient of the tangent** to **find the x-coordinate**. Next you will **substitute the x coordinate into the equation** of the curve to **find the y coordinate**. Lastly you will **substitute both the x and y coordinates into the linear equation** (straight line equation) and find* k.*

Step 1: Find the gradient of the tangent from the equation of normal.

$latex \displaystyle 3y+x=k$

$latex \displaystyle 3y=-x+k$

$latex \displaystyle y=-\frac{1}{3}x+\frac{k}{3}$

$latex \displaystyle {{m}_{2}}=-\frac{1}{3}$

$latex \displaystyle \text{Since}\ {{\text{m}}_{1}}\times {{m}_{2}}=-1$

$latex \displaystyle {{\text{m}}_{1}}=3$

Step 2: Differentiate the equation of the curve, equate it to the gradient of the tangent to find x.

$latex \displaystyle y={{x}^{2}}-5x+11$

$latex \displaystyle \frac{{dy}}{{dx}}=2x-5$

$latex \displaystyle 2x-5=3$

$latex \displaystyle 2x=8$

$latex \displaystyle x=4$

Step 3: Substitute the *x* value into the equation of the curve to find the y coordinate.

$latex \displaystyle y={{4}^{2}}-5(4)+11$

$latex \displaystyle y=7$

Step 4: Substitute both the *x* and *y* coordinate into the linear equation (straight line equation) to find *k.*

$latex \displaystyle 3y+x=k$

$latex \displaystyle 3(7)+4=k$

$latex \displaystyle k=25$

At first sight, the question look very simple. But if you look a little deeper, you realize that it is not that easy to find the x and y coordinates in order to find *k*. This is a typical exam of a O-Level exam question where you need to use formulas and methods from more then one topic to solve a problem. Some question will require knowledge from up to three different topics. So please don’t have a ‘tunnel vision’ and be flexible in using different methods and formulas.