HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Show that the following equation represents a pair of line. Find the acute angle between them:

(x - 3)² + (x - 3)(y - 4) - 2(y - 4)² = 0

Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x² + 2(sec 2α)xy + y² = 0

Find the condition that the equation ay² + bxy + ex + dy = 0 may represent a pair of lines. 

If the lines given by ax² + 2hxy + by² = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h².

If the line x + 2 = 0 coincides with one of the lines represented by the equation x² + 2xy + 4y + k = 0, then prove that k = - 4. 

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax² + 2hxy + by² = 0 is bx² - 2hxy + ay² = 0.

If equation ax² - y² + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

Find k if the slope of one of the lines given by 3x² + 4xy + ky² = 0 is three times the other.

Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`

Find the vector equation of the line passing through the point having position vector `-2hat"i" + hat"j" + hat"k"  "and parallel to vector"  4hat"i" - hat"j" + 2hat"k"`.

Find the vector equation of the line passing through points having position vector `3hati + 4hatj - 7hatk and 6hati - hatj + hatk`.

Find the vector equation of line passing through the point having position vector `5hat"i" + 4hat"j" + 3hat"k"` and having direction ratios  –3, 4, 2.

Find the vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k"  "and perpendicular to vectors"  hat"i" + hat"j" + hat"k" and 2hat"i" - hat"j" + hat"k"`.

Find the vector equation of the line passing through the point having position vector `-hat"i" - hat"j" + 2hat"k"  "and parallel to the line" bar"r" = (hat"i" + 2hat"j" + 3hat"k") + λ(3hat"i" + 2hat"j" + hat"k").`

Find the cartesian equations of the line passing through A(–1, 2, 1) and having direction ratios 2, 3, 1.

Find the Cartesian equations of the line passing through A(2, 2, 1) and B(1, 3, 0).

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.

Show that the lines given by `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1) and (x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` intersect. Also, find the coordinates of their point of intersection.

A line passes through (3, –1, 2) and is perpendicular to lines `bar"r" = (hat"i" + hat"j" - hat"k") + lambda(2hat"i" - 2hat"j" + hat"k") and bar"r" = (2hat"i" + hat"j" - 3hat"k") + mu(hat"i" - 2hat"j" + 2hat"k")`. Find its equation.

Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.