The line through the points (4, 3) and (2, 5) cuts off intercepts of l

1.	The line through the points (4, 3) and (2, 5) cuts off intercepts of lengths \(\lambda\) and \(\mu\) on the axes. Which one of the following is correct ?(a) \(\lambda\) > \(\mu\) (b) \(\lambda\) < \(\mu\) (c) \(\lambda\) > \(-\mu\) (d) \(\lambda\) = \(\mu\)
Answer» (d) \(\lambda\) = \(\mu\) Let the equation of the line in the intercept from be \(\frac{x}{\lambda}\)+ \(\frac{y}{\mu}\) = 1 Since it passes through (4, 3) and (2, 5) \(\frac{4}{\lambda}\) + \(\frac{3}{\mu}\) = 1 ....(i) and \(\frac{2}{\lambda}\) + \(\frac{5}{\mu}\) = 1 ....(ii) Multiplying (ii) by 2 and subtracting from (i), we get \(\bigg(\)\(\frac{4}{\lambda}\) + \(\frac{3}{\mu}\)\(\bigg)\) - \(\bigg(\)\(\frac{4}{\lambda}\) + \(\frac{10}{\mu}\)\(\bigg)\)= 1 - 2 ⇒ \(\frac{-7}{\mu}\) = -1⇒ \(\mu\) = 7 Putting \(\mu\) = 7 in (i), we get \(\frac{4}{\lambda}\) + \(\frac{3}{7}\) = 1 ⇒ \(\frac{4}{\lambda}\) = 1 - \(\frac{3}{7}\) = \(\frac{4}{7}\) = \(\lambda\) = 7 ∴ \(\lambda\) = \(\mu\) = 7.

The line through the points (4, 3) and (2, 5) cuts off intercepts of lengths \(\lambda\) and \(\mu\) on the axes. Which one of the following is correct ?(a) \(\lambda\) > \(\mu\) (b) \(\lambda\) < \(\mu\) (c) \(\lambda\) > \(-\mu\) (d) \(\lambda\) = \(\mu\)

Answer»

(d) \(\lambda\) = \(\mu\)

Let the equation of the line in the intercept from be

\(\frac{x}{\lambda}\)+ \(\frac{y}{\mu}\) = 1

Since it passes through (4, 3) and (2, 5)

\(\frac{4}{\lambda}\) + \(\frac{3}{\mu}\) = 1 ....(i)

and \(\frac{2}{\lambda}\) + \(\frac{5}{\mu}\) = 1 ....(ii)

Multiplying (ii) by 2 and subtracting from (i), we get

\(\bigg(\)\(\frac{4}{\lambda}\) + \(\frac{3}{\mu}\)\(\bigg)\) - \(\bigg(\)\(\frac{4}{\lambda}\) + \(\frac{10}{\mu}\)\(\bigg)\)= 1 - 2

⇒ \(\frac{-7}{\mu}\) = -1⇒ \(\mu\) = 7

Putting \(\mu\) = 7 in (i), we get \(\frac{4}{\lambda}\) + \(\frac{3}{7}\) = 1

⇒ \(\frac{4}{\lambda}\) = 1 - \(\frac{3}{7}\) = \(\frac{4}{7}\) = \(\lambda\) = 7

∴ \(\lambda\) = \(\mu\) = 7.