Question

Points $A$ and $B$ are in the first quadrant; point $O$ is the origin. If the slope of $OA$ is $1$,

slope of $OB$ is $7$ and $OA=OB$, then the slope of $AB$ is

• A. $-\frac{1}{5}$
• B. $\frac{-1}{4}$
• C. $\frac{-1}{3}$
• D. $\frac{-1}{2}$

Answer 1

Answer: (D)

$\frac{-1}{2}$

Let the point A(a,b) and B(c,d)

Slope of OA $=\frac{y_2-y_1}{x_2-x_1}=\frac{b-0}{a-0}=1$

So, $a=b$

Slope of OB $=\frac{y_2-y_1}{x_2-x_1}=\frac{d-0}{c-0}=7$

So, $d=7c$

Now,

$OA=OB$

$\sqrt{(a-0{)}^2+(b-0{)}^2}=\sqrt{(c-0{)}^2+(d-0{)}^2}$

Squaring on both sides

$a^2+b^2=c^2+d^2$

$a^2+a^2=c^2+(7c{)}^2$

$2a^2=50c^2$

$a^2=25c^2$

$a=5c$ [Only considering positive value since A and B is in the first quadrant]

Slope of AB $=\frac{d-b}{c-a}=\frac{7c-a}{c-a}$

$=\frac{7c-5c}{c-5c}$

$=\frac{2c}{-4c}$

$=-\frac{1}{2}$