Physics and Mathematics
Example – Intercept Form
Practice Questions with Step-by-Step Solutions
Question 1. Find the equation of the line which cuts off intercepts 5 on the x-axis and −3 on the y-axis.
Step-by-Step Solution:
x-intercept [a = 5]
y-intercept [b = −3]
Intercept form:
[\dfrac{x}{5} + \dfrac{y}{−3} = 1]
Simplify:
[\dfrac{x}{5} − \dfrac{y}{3} = 1]
Conclusion:
Equation of the line is [\dfrac{x}{5} − \dfrac{y}{3} = 1].
Question 2. Find the equation of the line whose intercepts on the coordinate axes are −4 and 6 respectively.
Step-by-Step Solution:
x-intercept [a = −4]
y-intercept [b = 6]
Intercept form:
[\dfrac{x}{−4} + \dfrac{y}{6} = 1]
Simplify:
[−\dfrac{x}{4} + \dfrac{y}{6} = 1]
Conclusion:
Equation of the line is [−\dfrac{x}{4} + \dfrac{y}{6} = 1].
Question 3. Find the equation of the line whose intercepts are in the ratio 1 : 2 and which cuts the y-axis at 8.
Step-by-Step Solution:
Let intercepts be [a = k], [b = 2k]
Given y-intercept [b = 8] ⇒ [2k = 8]
So, [k = 4]
Hence [a = 4], [b = 8]
Intercept form:
[\dfrac{x}{4} + \dfrac{y}{8} = 1]
Conclusion:
Equation of the line is [\dfrac{x}{4} + \dfrac{y}{8} = 1].
Question 4. Find the equation of the line which cuts equal intercepts on both axes and passes through the point (3, 1).
Step-by-Step Solution:
Let equal intercepts be [a = b]
Intercept form:
[\dfrac{x}{a} + \dfrac{y}{a} = 1]
Simplify:
[x + y = a]
Substitute point [(3, 1)]:
[3 + 1 = a]
So, [a = 4]
Conclusion:
Equation of the line is [x + y = 4].
Question 5. Find the equation of the line whose x-intercept is twice its y-intercept and which passes through (4, 3).
Step-by-Step Solution:
Let y-intercept [= k]
Then x-intercept [= 2k]
Intercept form:
[\dfrac{x}{2k} + \dfrac{y}{k} = 1]
Substitute point [(4, 3)]:
[\dfrac{4}{2k} + \dfrac{3}{k} = 1]
Simplify:
[\dfrac{2 + 3}{k} = 1]
So, [k = 5]
Conclusion:
Equation of the line is [\dfrac{x}{10} + \dfrac{y}{5} = 1].
Question 6. Find the equation of the line whose intercepts are −a and a.
Step-by-Step Solution:
x-intercept [= −a]
y-intercept [= a]
Intercept form:
[\dfrac{x}{−a} + \dfrac{y}{a} = 1]
Simplify:
[y − x = a]
Conclusion:
Equation of the line is [y − x = a].
Question 7. Find the equation of the line whose intercepts are 3 times and 2 times the respective intercepts of the line [\dfrac{x}{2} + \dfrac{y}{3} = 1].
Step-by-Step Solution:
Given intercepts: [a = 2], [b = 3]
New x-intercept [= 3 × 2 = 6]
New y-intercept [= 2 × 3 = 6]
Intercept form:
[\dfrac{x}{6} + \dfrac{y}{6} = 1]
Simplify:
[x + y = 6]
Conclusion:
Equation of the required line is [x + y = 6].
Question 8. Find the equation of the line which cuts off intercepts whose sum is 10 and which cuts the x-axis at twice the y-intercept.
Step-by-Step Solution:
Let y-intercept [= k]
Then x-intercept [= 2k]
Given sum of intercepts:
[2k + k = 10]
So, [k = \dfrac{10}{3}]
Hence [a = \dfrac{20}{3}], [b = \dfrac{10}{3}]
Intercept form:
[\dfrac{x}{20/3} + \dfrac{y}{10/3} = 1]
Conclusion:
Equation of the line is [\dfrac{3x}{20} + \dfrac{3y}{10} = 1].
Question 9. Find the equation of the line whose intercepts are reciprocals of each other and which cuts the y-axis at 2.
Step-by-Step Solution:
y-intercept [b = 2]
x-intercept is reciprocal ⇒ [a = \dfrac{1}{2}]
Intercept form:
[\dfrac{x}{1/2} + \dfrac{y}{2} = 1]
Simplify:
[2x + \dfrac{y}{2} = 1]
Conclusion:
Equation of the line is [2x + \dfrac{y}{2} = 1].
Question 10. Find the equation of the line which cuts intercepts on the axes in the ratio 3 : −2 and passes through (6, 1).
Step-by-Step Solution:
Let x-intercept [= 3k], y-intercept [= −2k]
Intercept form:
[\dfrac{x}{3k} + \dfrac{y}{−2k} = 1]
Substitute point [(6, 1)]:
[\dfrac{6}{3k} − \dfrac{1}{2k} = 1]
Simplify:
[\dfrac{4 − 1}{2k} = 1]
So, [k = \dfrac{3}{2}]
Conclusion:
Equation of the line is [\dfrac{x}{9/2} − \dfrac{y}{3} = 1].
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