Questions: Straight Lines
Straight Lines24 min readLesson 5 of 6
Q1. Find the slope of the line passing through the points (2, 3) and (5, 9).
Answer: The slope is the change in y divided by the change in x: (9 minus 3) by (5 minus 2), which is 6 by 3 equal to 2. The line rises 6 units while moving 3 units to the right, so the slope is positive, as expected.
Hinglish: Slope = (9 − 3)/(5 − 2) = 6/3 = 2. Line right ki taraf jaate hue upar uthti hai, isliye slope positive hai.
Q2. Find the inclination of the line whose slope is 1.
Answer: The tangent of the inclination is 1, and the inclination lies between 0 and 180 degrees. The angle whose tangent is 1 is 45 degrees, so the inclination is 45 degrees.
Hinglish: tan θ = 1 se θ = 45°, kyunki inclination 0° se 180° ke beech hoti hai.
Q3. A line has slope 3 and passes through (1, 2). Write its equation.
Answer: The slope and one point are both given, so the point-slope form applies at once: y minus 2 equals 3 times (x minus 1). Expanding gives y equal to 3x minus 1, which is the slope-intercept form. As a check, substituting x equal to 1 gives y equal to 2, so the point lies on the line.
Hinglish: Point-slope form: y − 2 = 3(x − 1) → y = 3x − 1. x = 1 daalne par y = 2 aata hai, point satisfy hota hai.
Q4. Find the equation of the line with x-intercept 4 and y-intercept 3.
Answer: Use the intercept form: x by 4 plus y by 3 equal to 1. Multiplying by 12 gives 3x plus 4y equal to 12, which is the general form. The slope is minus 3 by 4 and both intercepts are positive, matching a line that falls from the y-axis to the x-axis.
Hinglish: Intercept form: x/4 + y/3 = 1 → 3x + 4y = 12. Slope −3/4 hai aur dono intercepts positive hain.
Q5. Find the angle between the lines with slopes 2 and minus 1 by 2.
Answer: The product of the slopes is 2 times minus 1 by 2, which is minus 1, so the lines are perpendicular and the angle between them is 90 degrees. The tangent formula confirms this because the denominator 1 plus m1m2 becomes zero.
Hinglish: m₁m₂ = 2 × (−1/2) = −1, isliye lines perpendicular hain aur angle 90° hai. Formula me denominator 1 + m₁m₂ zero ho jaata hai.
Q6. Work out the perpendicular distance from the point (2, 3) to the line 3x plus 4y plus 9 equal to 0.
Answer: Put the coordinates into the distance formula: the numerator is 3 times 2 plus 4 times 3 plus 9, divided by the square root of 9 plus 16. The numerator is 27 and the denominator is 5, so the distance is 27 by 5, or 5.4 units.
Hinglish: |3(2) + 4(3) + 9| / √(9+16) = 27/5 = 5.4 units. Denominator √(A²+B²) hai, A+B nahi.
Q7. Two parallel lines are 3x minus 4y plus 9 equal to 0 and 6x minus 8y minus 15 equal to 0. How far apart are they?
Answer: Halve the second equation so the coefficients match: 3x minus 4y minus 15 by 2 equal to 0. The constants are 9 and minus 15 by 2, whose difference is 33 by 2. The denominator is 5, so the distance is 33 by 10.
Hinglish: Dusri line ko 2 se divide karo: 3x − 4y − 15/2 = 0. |9 − (−15/2)| = 33/2, denominator 5, distance = 33/10.
Q8. Find the slope of a line perpendicular to the line 4x minus 3y plus 7 equal to 0.
Answer: The slope of the given line is minus A by B, which is minus 4 by minus 3, that is 4 by 3. The perpendicular slope is the negative reciprocal, which is minus 3 by 4.
Hinglish: Di hui line ka slope 4/3 hai, isliye perpendicular ka slope −3/4 hoga (negative reciprocal).
Q9. Check whether the points (1, 2), (3, 6) and (5, 10) are collinear.
Answer: Find the slopes of the first two pairs. Between (1, 2) and (3, 6) the slope is 4 by 2 equal to 2. Between (3, 6) and (5, 10) the slope is 4 by 2 equal to 2. The slopes match, so the three points lie on one line with equation y equal to 2x.
Hinglish: Dono pairs ka slope 2 aata hai, isliye points collinear hain. Line ki equation y = 2x hai, teeno points ise satisfy karte hain.
Q10. A line passes through (2, 1) and is parallel to the line 2x minus 3y plus 4 equal to 0. Find its equation.
Answer: The given line has slope 2 by 3, so the parallel line also has slope 2 by 3. Using the point-slope form through (2, 1): y minus 1 equals 2 by 3 times (x minus 2). Multiplying by 3 gives 2x minus 3y minus 1 equal to 0, which is the required equation in general form.
Hinglish: Slope 2/3 (parallel line ka same slope). Point-slope: y − 1 = (2/3)(x − 2) → 2x − 3y − 1 = 0.
One more habit for the questions above: after finding the equation of a line, rewrite it in general form and check that the coefficients are integers with no common factor left. This is the form the answer options use, and it makes comparison immediate.
Two extra checks for the questions above. A line through two given points can be tested by substituting both points back into the final equation, and a distance answer can be tested by drawing a rough sketch and comparing the size of the answer with the coordinates. Both checks take seconds and catch most arithmetic slips.
Common mistakes on this chapter
Mixing the order of subtraction in the slope formula flips the sign. Using the slope of a line instead of its negative reciprocal in a perpendicular question gives a line that crosses at the wrong angle. Forgetting the absolute value in the distance formula gives a negative distance, which should immediately look wrong. Applying the parallel-line distance formula before matching the coefficients of x and y gives a wrong denominator. Check the answer against these four points before moving on, and keep a small sketch beside the algebra for every second question.
Quiz
The quiz below has five multiple choice questions on slopes, angles, line forms and distances. Attempt it after working through the questions above, then move to the previous year page.