Q27 of 45 Page 1

Find the vector equation of the plane passing through three points with position vectors and Also, find the coordinates of the point of intersection of this plane and the line

Let the position vectors of the three points be



So, the equation of the plane passing through the points is







Now, we know that, In cross product









…(i)


So, the vector equation of the required plane is


The equation of the given line is


Position vector of any point on the give line is


…(ii)


The point (ii) lies on plane (i) if,



⇒ 9(3 + 2λ) + 3(-1 - 2λ) – 1(-1 + λ) = 14


⇒ 27 + 18λ – 3 - 6λ + 1 – λ = 14


⇒ 11λ + 25 = 14


⇒ 11λ = 14 – 25


⇒ 11λ = -11


⇒ λ = -1


Putting λ = -1 in eq. (ii), we get



⇒


⇒


Thus, the position vector of the point of intersection of the given line and the plane (i) is and its coordinates are (1, 1, -2)


More from this chapter

All 45 →
25

Using integration, find the area bounded by the curve x2 = 4y and the line x = 4y - 2.

OR


Using integration, find the area of the region enclosed between the two circles x2 + y2 = 4 and (x - 2)2 + y2 = 4.

 26

Show that the differential equation 2yex/y dx + (y - 2xex/y)dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.

 28

A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as ₹10,500 and ₹9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

 29

Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.