Question
If $\text{a}\neq\text{b}\neq\text{c},$ prove that the points $(a, a^2), (b, b^2), (c, c^2)$ can never be collinear.
✓
Answer
Let $(a, a^2), (b, b^2), (c, c^2)$ he given points. Three points are collinear if area enclosed by three points is zero. Area of $\triangle\text{ABC}=\frac{1}{2}|\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)|$ $=\frac{1}{2}|\text{a}(\text{b}^2-\text{c}^2)+\text{b}(\text{c}^2-\text{a}^2)+\text{c}(\text{a}^2-\text{b}^2)|$ $=\frac{1}{2}|\text{ab}^2-\text{ac}^2+\text{bc}^2-\text{a}^2\text{b}+\text{a}^2\text{c}-\text{b}^2\text{c}|$ $=\frac{1}{2}|(\text{a}^2\text{c}-\text{a}^2\text{b})+(\text{ab}^2-\text{ac}^2)+(\text{bc}^2-\text{b}^2\text{c})|$ $=\frac{1}{2}|(-\text{a}^2)(\text{b}-\text{c})+\text{a}(\text{b}^2-\text{c}^2)-\text{bc}(\text{b}-\text{c})|$ $=\frac{1}{2}|(\text{b}-\text{c})(-\text{a}^2+\text{a}(\text{b}+\text{c})-\text{bc})|$$=\frac{1}{2}|(\text{b}-\text{c})(-\text{a}^2+\text{ab}+\text{ac}-\text{bc})|$
$=\frac{1}{2}|(\text{b}-\text{c})[(-\text{a})(\text{a}-\text{b})+\text{c}(\text{a}-\text{b})]|$
$=\frac{1}{2}|(\text{b}-\text{c})(\text{c}-\text{a})(\text{a}-\text{b})|$
It is given that $\text{a}\neq\text{b}\neq\text{c}$ Hence area of triangle made by three points is never zero. Hence given points are never collinear.
$=\frac{1}{2}|(\text{b}-\text{c})[(-\text{a})(\text{a}-\text{b})+\text{c}(\text{a}-\text{b})]|$
$=\frac{1}{2}|(\text{b}-\text{c})(\text{c}-\text{a})(\text{a}-\text{b})|$
It is given that $\text{a}\neq\text{b}\neq\text{c}$ Hence area of triangle made by three points is never zero. Hence given points are never collinear.
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 P(9a - 2, -b) divides the line segment joining A(3a + 1, -3) and B(8a, 5) in the ratio 3 : 1, find the values of a and b.
A(6, 1), B(8, 2) and C(9, 4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of $\triangle\text{ADE}.$
→For the following statments state whether true (T) or false(F):If O is any point inside a rectangle ABCD then $OA^2 + OC^2 = OB^2 + OD^2$
→A metallic bucket, open at the top, of height 24cm is in the form of the frustum of a cone, the radii of whose lower and upper circular ends are 7cm and 14cm, respectively. Find
- The volume of water which can completely fill the bucket,
- The area of the metal sheet used to make the bucket.
From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot of the tower are α and $\beta(\beta>\alpha).$ Find the distance between the two objects.
→Find all the zeros of the polynomial $(2x^4 - 11x^3 + 7x^2 + 13x)$, it being given that two if its zeros are $3+\sqrt2$ and $3-\sqrt2.$
→Find the distance between the points:
$\text{P}(\text{a}\sin\alpha,\text{a}\cos\alpha)$ and $\text{Q}(\text{a}\cos\alpha, -\text{a}\sin\alpha)$
→$\text{P}(\text{a}\sin\alpha,\text{a}\cos\alpha)$ and $\text{Q}(\text{a}\cos\alpha, -\text{a}\sin\alpha)$
The first term of an AP is p and its common difference is q. Find its $10^{th}$ term.
→If the sum of first $6$ terms of an $A.P$. is $36$ and that of the first $16$ terms is $256,$ find the sum of first $10$ terms.
→A wooden article was made by scooping out a hemisphere from each end of a solid cylinder as shown in the figure. If the height of the cylinder is $10 \ cm$ and its base is of radius $3.5 \ cm,$ find the total surface area of the article.
→