Question

Find the value of k, if the lines joining the origin to the points of intersection of the curve
2x² - 2xy+3y²+ 2x-y-1=0 and the line x +2y = k are mutually perpendicular.
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Answer 1

Answer:

Straight lines is an extremely important topic of IIT JEE Mathematics. It often fetches some direct questions in various competitions like the IIT JEE. Since the topic is quite vast, students are advised to spend sufficient time on grasping the various concepts. Angle between pair of straight lines is an important head under straight lines. We begin with the concept of angle between pair of lines and then discuss some of the illustrations on the same:

Angle between two Straight Lines

Suppose we have two straight lines y = m1x + c1 and y = m2x + c2, then the angle between these two lines is given by tan θ = |(m1 – m2)/ (1 + m1m2)|.

If m1, m2 and m3 are the slopes of three lines L1 = 0, L2 = 0 and L3 = 0, where m1 > m2 > m3 then the interior angles of the triangle ABC formed by these lines are given by,

tan A = (m1 - m2)/(1 + m1m2), tan B = (m2 - m3)/(1 + m2m3) and tan C = (m3 - m1)/(1 + m3m1).

Remark:

If one of the line is parallel to y-axis then the angle between two straight lines is given by tan θ = ±1/m where ‘m’ is the slope of the other straight line.

If the two lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the formula becomes tan θ = |(a1b2 - b1a2)/(a1a2 + b1b2)|

Generally speaking, the angle between these two lines is assumed to be acute and hence, the value of tan θ is taken to be positive.

Result:

Angle between pair of lines represented by ax2 + 2hxy + by2 = 0

Comparing the coefficients of x2, y2 and xy, we get

b(y – m1x) (y – m2x) = ax2 + 2hxy + by2

m1 + m2 = –2h/b and

m1 m2 = a/b

tan θacute = |(m2 – m1)/(1 + m1m2)|

= |√(m1–m2)2 – 4m1m2/(1 + m1m2)|

= |2√(h2 – ab)/(a + b)|

Step-by-step explanation:

Hopefully it's help you.........

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