MCQ
If points $(t, 2t), (-2, 6)$ and $(3, 1)$ are collinear, then $t =$
- A$\frac{3}{4}$
- ✓$\frac{4}{3}$
- C$\frac{5}{3}$
- D$\frac{3}{5}$
✓
Answer
Correct option: B.
$\frac{4}{3}$
We have three collinear points $A(t, 2t), B(-2, 6), C(3, 1).$
In general if $A\left(x_1, y_1\right), B\left(x_2, y_2\right), C\left(x_3, y_3\right)$ are collinear then,
$\frac{1}{2}\Big[\text{x}_1(\text{y}_2 - \text{y}_3) + \text{x}_2(\text{y}_3 - \text{y}_1) + \text{x}_3\text{(y}_1 - \text{y}_2)\Big] = 0$
So,
$t(6 - 1) - 2(1 - 2t) + 3(2t - 6) = 0$
So,
$5t + 4t + 6t - 2 - 18 = 0$
So,
$15t = 20$
Therefore,
$\text{t}=\frac{4}{3}$
In general if $A\left(x_1, y_1\right), B\left(x_2, y_2\right), C\left(x_3, y_3\right)$ are collinear then,
$\frac{1}{2}\Big[\text{x}_1(\text{y}_2 - \text{y}_3) + \text{x}_2(\text{y}_3 - \text{y}_1) + \text{x}_3\text{(y}_1 - \text{y}_2)\Big] = 0$
So,
$t(6 - 1) - 2(1 - 2t) + 3(2t - 6) = 0$
So,
$5t + 4t + 6t - 2 - 18 = 0$
So,
$15t = 20$
Therefore,
$\text{t}=\frac{4}{3}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If the point $P (6,2)$ divides the line segment joining $A(6,5)$ and $B(4, y)$ in the ratio $3: 1$, then the value of $y$ is
→The height of a cone is $30\ cm$. A small cone is cut off at the top by a plane parallel to the base. If its volume be of the volume of the given cone, then the height above the base at which the section has been made, is:
→Choose the correct answer from the given four options:
If the $2^{\text {nd }}$ term of an $AP$ is $13$ and the $5^{\text {th }}$ term is $25$ , what is its $7^{\text {th }}$ term?
→If the $2^{\text {nd }}$ term of an $AP$ is $13$ and the $5^{\text {th }}$ term is $25$ , what is its $7^{\text {th }}$ term?
Two dice are thrown together. The probability that they show different numbers is :
If $\triangle\text{ABC} \sim \triangle\text{DEF}$ such that $AB = 1.2\ cm$ and $DE = 1.4\ cm,$ the ratio of the areas of $\triangle\text{ABC}$ and $\triangle\text{DEF}$ is :
→$3\cos^260^\circ+2\cot^230^\circ-5\sin^245^\circ=?$
→Find the least positive integer divisible by 20 and 24.
The first three terms of an $AP$ respectively are $3 y-1,3 y+5$ and $5 y+1$. Then $y$ equals:
→If the radius of a semi-circular protractor is 7 cm , then its perimeter is:
Two circles touch internally at point $Q$. From an external point $R$, two tangents $R M$ and $R N$ are drawn to the two circles. Then,
→