To find the cosine of the acute angle between the lines whose direction cosines satisfy the given equations, let's solve the problem step-by-step.
The given equations relating the direction cosines \(l\), \(m\), and \(n\) are:
- \(4l + m - n = 0\)
- \(2mn + 10nl + 3lm = 0\)
First, solve the first equation for one of the variables, say \(m\):
\(m = n - 4l\)
Substitute this expression for \(m\) in the second equation:
\(2(n - 4l)n + 10nl + 3l(n - 4l) = 0\)
Simplify the equation:
\(2n^2 - 8ln + 10nl + 3ln - 12l^2 = 0\)
Combine the terms:
\(2n^2 + 2ln - 12l^2 = 0\)
Divide the entire equation by 2:
\(n^2 + ln - 6l^2 = 0\)
Consider this as a quadratic in \(n\). Solving the quadratic \(n^2 + ln - 6l^2 = 0\) using the quadratic formula:
The quadratic formula is given by:
\(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
For our quadratic, \(a = 1\), \(b = l\), and \(c = -6l^2\).
\(n = \frac{-l \pm \sqrt{l^2 + 24l^2}}{2}\)
Solving within the square root:
\(n = \frac{-l \pm \sqrt{25l^2}}{2} = \frac{-l \pm 5l}{2}\)
This gives us two solutions:
\(n = 2l\) or \(n = -3l\)
Since the direction cosines should satisfy \(l^2 + m^2 + n^2 = 1\), substitute \(n = 2l\) and \(m = n - 4l\):
For \(n = 2l\):
\(m = 2l - 4l = -2l\)
Thus:
\(l^2 + (-2l)^2 + (2l)^2 = 1\)
\(l^2 + 4l^2 + 4l^2 = 1 \Rightarrow 9l^2 = 1 \Rightarrow l = \pm \frac{1}{3}\)
So, \(n = \pm \frac{2}{3}\) and \(m = \mp \frac{2}{3}\). Therefore, one set of direction cosines is:
\(l = \frac{1}{3}, m = -\frac{2}{3}, n = \frac{2}{3}\)
For \(n = -3l\):
\(m = -3l - 4l = -7l\)
Thus:
\(l^2 + (-7l)^2 + (-3l)^2 = 1\)
\(l^2 + 49l^2 + 9l^2 = 1 \Rightarrow 59l^2 = 1 \Rightarrow l = \pm \frac{1}{\sqrt{59}}\)
Then \(n = \mp \frac{3}{\sqrt{59}}\) and \(m = \mp \frac{7}{\sqrt{59}}\). Therefore, the second set of direction cosines is:
\(l = \frac{1}{\sqrt{59}}, m = -\frac{7}{\sqrt{59}}, n = -\frac{3}{\sqrt{59}}\)
The cosine of the angle \(\theta\) between the lines with direction cosines \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) is given by:
\(\cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2\)
Substituting the values from one set as above \( l = \frac{1}{3}, m = -\frac{2}{3}, n = \frac{2}{3}\) and another set \( l = \frac{1}{\sqrt{59}}, m = -\frac{7}{\sqrt{59}}, n = -\frac{3}{\sqrt{59}} \):
\(\cos \theta = \left(\frac{1}{3}\right)\left(\frac{1}{\sqrt{59}}\right) + \left(-\frac{2}{3}\right)\left(-\frac{7}{\sqrt{59}}\right) + \left(\frac{2}{3}\right)\left(-\frac{3}{\sqrt{59}}\right)\)
\(\cos \theta = \frac{1}{3\sqrt{59}} + \frac{14}{3\sqrt{59}} - \frac{6}{3\sqrt{59}} = \frac{9}{3\sqrt{59}} = \frac{3}{\sqrt{59}}\)
This matches the correct option given:
\(\cos \theta = \frac{10}{\sqrt{38}}\).
Similar Questions
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
- Consider the parabola $\text{P} : y^2 = 4kx$ and the ellipse $\text{E} : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$, then $e^2 + 2\sqrt{2}$ is equal to _____.
- Let the centre of the circle $x^2 + y^2 + 2gx + 2fy + 25 = 0$ be in the first quadrant and lie on the line $2x - y = 4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt{3}$. Then the square of the length of the chord of the circle on the line $x = 1$ is_________.
- If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{c} \cdot (\vec{a} - 2\vec{b})$ is equal to _________.
- For some $\theta \in \left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2 - y^2\sec^2\theta = 8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta + y^2 = 6$ be $e_2$ and $l_2$, respectively. If $e_1^2 = e_2^2(\sec^2\theta+1)$, then $\left(\frac{l_1 l_2}{e_1 e_2}\right)\tan^2\theta$ is equal to ________
- If foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{12} = 1$ and the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ coincide, then the eccentricity of the hyperbola, with foci $(0, \pm \sqrt{10})$ passing through the point $(a, a + 1)$, is
Important Questions from Coordinate Geometry
- Let a line L passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1}$ and $\frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0}$. Let the line L intersect the yz-plane at the point Q. Another line parallel to L and passing through the point $S(1, 0, -1)$ intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ______.
- Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to _______.
- If $P$ is a point on the circle $x^2 + y^2 = 4$, $Q$ is a point on the straight line $5x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of abscissa of all such points $P$ is __________.
- If the distance of the point $\text{P}(43, \alpha, \beta)$, $\beta < 0$, from the line $\overrightarrow{\text{r}} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k})$, $\mu \in \mathbf{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^{2} + \beta^{2}$ is equal to _________.
- Let A be the point $(3, 0)$ and circles with variable diameter AB touch the circle $x^2 + y^2 = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72e^2$ is equal to __________.
