Download App

Straight Lines — Applied Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceApplied MathsStraight Lines5 Marks
Question
If three lines whose equations are $y=m_1 x+c_1$ $y=m_2 x+c_2$ and $y=m_3 x+c_3$ are concurrent, then show that $m_1\left(c_2-c_3\right)+m_2\left(c_3-c_1\right)+m_3\left(c_1-c_2\right)=0$.
✓

Answer

The equation of the given lines are
$y=m_1 x+c_1\ldots(i)$
$\begin{array}{l}
y=m_2 x+c_2\ldots(ii) \\
y=m_3 x+c_3\ldots(iii)
\end{array}$
On subtracting equation (i) from (ii), we obtain
0 = (m2 - m1 x + (c2 - c1)
⇒(m1 - m2)x = c2 - c1
⇒$x=\frac{c_2-c_1}{m_1-m_2}$
On substituting this value of x in (i), we obtain
$y=m_1\left(\frac{c_2-c_1}{m_1-m_2}\right)+c_1$
$y=\frac{m_1 c_2-m_1 c_1}{m_1-m_2}+c_1$
$y=\frac{m_1 c_2-m_1 c_1+m_1 c_1-m_2 c_1}{m_1-m_2}$
$y=\frac{m_1 c_2-m_2 c_1}{m_1-m_2}$
$\therefore \quad\left(\frac{c_2-c_1}{m_1-m_2}, \frac{m_1 c_2-m_2 c_1}{m_1-m_2}\right)$ is the point of intersection of lines (i) and (ii).
It is given that lines (i), (ii) and (iii) are concurrent. Hence, the point of intersection of lines (i) and (ii) will also satisfy equation (iii),
$\frac{m_1 c_2-m_2 c_1}{m_1-m_2}=m_3\left(\frac{c_2-c_1}{m_1-m_2},\right)+c_3$
$\frac{m_1 c_2-m_2 c_1}{m_1-m_2}=\frac{m_3 c_2-m_3 c_1+c_3 m_1-c_3 m_2}{m_1-m_2}$
$\begin{aligned} m_1 c_2-m_2 c_1-m_3 c_2+m_3 c_1-c_3 m_1+c_3 m_2 & =0 \\ m_1\left(c_2-c_3\right)+m_2\left(c_3-c_1\right)+m_3\left(c_1-c_2\right) & =0\end{aligned}$
Hence, $m_1\left(c_2-c_3\right)+m_2\left(c_3-c_1\right)+m_3\left(c_1-c_2\right)=0$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Straight Lines→Straight Lines — all question types→Applied Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$
→
Find the sum of first ' $n$ ' terms of the series $0.7+$ $0.77+0.777+\ldots$
→
(a) ₹ 200 is invested at the end of each month in an account paying interest 6% per year compounded monthly. What is the future value of this annuity after 10th payment? Given that (1.005)10 = 1.0511 deg
(b) Find the present value of ₹ 10,000 to be required after 5 years if the interest rate be 9%. Given that (1.09)5 = 1.5386
(c) Sneha borrows ₹ 500000 to buy a house. If she pays equal installments for 20 years and 10% interest on outstanding balance what will be the equal annual installment ? Given that ( 1.10)20 =6.7274
→
Find the mean deviation about the median for the following data:
xi3691213152122
$f_i$34524543
→
A shopkeeper buys an article whose list price is ₹ 8000 at some rate of discount from a wholesaler. He sells the article to a consumer at the list price. The sales are intra-state and the rate of GST is $18 \%$. If the shopkeeper pay a tax (under GST) of ₹72 to the State Government, find the rate of discount at which he bought the article from the wholesaler.
→
Find the sum of the series, $1.3 .4+5.7 .8+9.11 .12+\ldots$ upto $n$ terms.
→
How many 3-digit numbers are there which have exactly one of their digits as 6 ?
→
For the following distribution, find central moments, $\beta_1$ and $\beta_2$.
Class1.5 - 2.52.5 - 3.53.5 - 4.54.5 - 5.55.5 - 6.5
Frequency13731
→
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that one of them is black and other is red.
→
Three coins are tossed simultaneously. Consider the event $E$ 'three heads or three tails,' $F$ 'at least two heads' and $G$ 'at most two heads'. Of the pairs $(E, F),(E, G)$ and $(F, G)$, which are independent ? Which are dependent?
→