A line with equation $$\(y=m x-6\)$$ is a tangent to the curve with equation $$\(y=x^{2}-4 x+3\)$$. Find the possible values of the constant $$\(m\)$$, and the corresponding coordinates of the points at which the line touches the curve. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_s21_qp_13 Year：2021 Question No：3
Answer:
$x^{2}-4 x+3=m x-6$ leading to $x^{2}-x(4+m)+9$
$b^{2}-4 a c$ leading to $(4+m)^{2}-4 \times 9$
$4+m=\pm 6$ or $(m-2)(m+10)=0$ leading to $m=2$ or $-10$
Substitute both their $m$ values into their equation in line 1 $m=2$
leading to $x=3 ; m=-10$ leading to $x=-3$
$(3,0),(-3,24)$
Alternative method for Question 3
$\frac{d y}{d x}=2 x-4 \rightarrow 2 x-4=m$
$x^{2}-4 x+3=(2 x-4) x-6$
$x^{2}-4 x+3=2 x^{2}-4 x-6 \rightarrow 9=x^{2} \rightarrow x=\pm 3$
$y=0,24$ or $(3,0),(-3,24)$
Substitute both their $x$ values into their equation in line 1
When $x=3, m=2 ;$ when $x=-3, m=-10$
$b^{2}-4 a c$ leading to $(4+m)^{2}-4 \times 9$
$4+m=\pm 6$ or $(m-2)(m+10)=0$ leading to $m=2$ or $-10$
Substitute both their $m$ values into their equation in line 1 $m=2$
leading to $x=3 ; m=-10$ leading to $x=-3$
$(3,0),(-3,24)$
Alternative method for Question 3
$\frac{d y}{d x}=2 x-4 \rightarrow 2 x-4=m$
$x^{2}-4 x+3=(2 x-4) x-6$
$x^{2}-4 x+3=2 x^{2}-4 x-6 \rightarrow 9=x^{2} \rightarrow x=\pm 3$
$y=0,24$ or $(3,0),(-3,24)$
Substitute both their $x$ values into their equation in line 1
When $x=3, m=2 ;$ when $x=-3, m=-10$
Knowledge points:
1.1.2 find the discriminant of a quadratic polynomial and use the discriminant
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.1.4 solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic
1.3.5 understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. (e.g. to determine the set of values of for which the line intersects, touches or does not meet a quadratic curve.)
Solution:
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