Advertisements
Advertisements
Question
Find the values of α so that the point P (α2, α) lies inside or on the triangle formed by the lines x − 5y+ 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0.
Solution
Let ABC be the triangle of sides AB, BC and CA whose equations are x − 5y + 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0, respectively.
On solving the equations, we get A (9,3), B (4, 2) and C (13, 5) as the coordinates of the vertices.
It is given that point P (α2, α) lies either inside or on the triangle. The three conditions are given below.
(i) A and P must lie on the same side of BC.
(ii) B and P must lie on the same side of AC.
(iii) C and P must lie on the same side of AB.
If A and P lie on the same side of BC, then
\[\left( 9 - 9 + 2 \right)\left( \alpha^2 - 3\alpha + 2 \right) \geq 0\]
\[ \Rightarrow \left( \alpha - 2 \right)\left( \alpha - 1 \right) \geq 0\]
\[\Rightarrow \alpha \in ( - \infty , 1] \cup [2, \infty )\] ... (1)
If B and P lie on the same side of AC, then
\[\left( 4 - 4 - 3 \right)\left( \alpha^2 - 2\alpha - 3 \right) \geq 0\]
\[ \Rightarrow \left( \alpha - 3 \right)\left( \alpha + 1 \right) \leq 0\]
\[\Rightarrow \alpha \in \left[ - 1, 3 \right]\] ... (2)
If C and P lie on the same side of AB, then
\[\left( 13 - 25 + 6 \right)\left( \alpha^2 - 5\alpha + 6 \right) \geq 0\]
\[ \Rightarrow \left( \alpha - 3 \right)\left( \alpha - 2 \right) \leq 0\]
\[\Rightarrow \alpha \in \left[ 2, 3 \right]\] ... (3)
From (1), (2) and (3), we get: \[\alpha \in \left[ 2, 3 \right]\]
APPEARS IN
RELATED QUESTIONS
Reduce the following equation into intercept form and find their intercepts on the axes.
3y + 2 = 0
Find equation of the line perpendicular to the line x – 7y + 5 = 0 and having x intercept 3.
Prove that the line through the point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x –x1) + B (y – y1) = 0.
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c.
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and xsec θ+ y cosec θ = k, respectively, prove that p2 + 4q2 = k2.
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that `1/p^2 = 1/a^2 + 1/b^2`.
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.
Find the equation of the bisector of angle A of the triangle whose vertices are A (4, 3), B (0, 0) and C(2, 3).
Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y =1.
Find the equation of a line for p = 5, α = 60°.
Find the equation of a line for p = 8, α = 225°.
The length of the perpendicular from the origin to a line is 7 and the line makes an angle of 150° with the positive direction of Y-axis. Find the equation of the line.
Find the equation of a straight line on which the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area \[50/\sqrt{3}\] with the axes.
Reduce the following equation to the normal form and find p and α in y − 2 = 0.
Put the equation \[\frac{x}{a} + \frac{y}{b} = 1\] to the slope intercept form and find its slope and y-intercept.
Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].
Find the point of intersection of the following pairs of lines:
2x − y + 3 = 0 and x + y − 5 = 0
Find the coordinates of the vertices of a triangle, the equations of whose sides are x + y − 4 = 0, 2x − y + 3 = 0 and x − 3y + 2 = 0.
Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.
Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).
Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
Determine whether the point (−3, 2) lies inside or outside the triangle whose sides are given by the equations x + y − 4 = 0, 3x − 7y + 8 = 0, 4x − y − 31 = 0 .
Write the coordinates of the orthocentre of the triangle formed by the lines xy = 0 and x + y = 1.
If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in
The figure formed by the lines ax ± by ± c = 0 is
Two vertices of a triangle are (−2, −1) and (3, 2) and third vertex lies on the line x + y = 5. If the area of the triangle is 4 square units, then the third vertex is
Find the equation of the line where length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
The inclination of the line x – y + 3 = 0 with the positive direction of x-axis is ______.
Find the equation of the line which passes through the point (– 4, 3) and the portion of the line intercepted between the axes is divided internally in the ratio 5 : 3 by this point.
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
6x + 3y – 5 = 0
Reduce the following equation into intercept form and find their intercepts on the axes.
4x – 3y = 6
